GCSE Maths: Tree diagrams: with and without replacement (Foundation)
Part of the GCSE Maths probability chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Tree diagrams: with and without replacement
A tree diagram is a picture of what can happen in two or more events, one after the other. Each stage has its own set of branches. On each branch we write the probability of that result, and at the end of each complete route we write the combined outcome. Here is a bag with 3 red and 2 blue counters. One counter is picked, its colour is noted and it is put back. Then a second counter is picked.
Rules for tree diagrams. (1) The probabilities on the branches leaving any one point add up to 1, because one of those things must happen. (2) To find the probability of a complete route, multiply along the branches (the “and” rule: first thing and second thing). (3) To find the probability of an event that can happen by more than one route, add the probabilities of those routes (the “or” rule: this route or that route). (4) The probabilities on all the complete routes add up to 1, which is a very good check.
Without replacement. What if the first counter is not put back? Then there are only 4 counters left for the second pick, and the numbers on the second branches change. If the first counter was red, 2 red and 2 blue are left, so the second branches are 2/4 and 2/4. If the first was blue, 3 red and 1 blue are left, so they are 3/4 and 1/4. The two picks are now dependent: the first result changes the probabilities for the second. This is the key difference: with replacement the branches are the same each time; without replacement the bottom number falls by one and the top number for the colour just taken falls by one.
Not just bags. Trees work for any events: wet or dry days, a free throw that is scored or missed, a test that is passed or failed. If the probabilities on the second stage depend on the first result, draw different second branches for each first branch. And for “at least one”, remember the shortcut: find the probability of the opposite (none) and subtract from 1.
Key ideas
- Branch probabilities from one point add up to 1.
- Multiply along the branches (and). Add the routes you want (or).
- The probabilities of all the complete routes add up to 1.
- With replacement: the same branches at each stage (independent). Without replacement: the numbers change (dependent).
- For “at least one”, use 1 − P(none).
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 35 × 35 = 925
Check. The routes not used add up to 1625, and 925 + 1625 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then blue
blue then red - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then blue: 35 × 25 = 625
blue then red: 25 × 35 = 625 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.625 + 625 = 1225
Check. The routes not used add up to 1325, and 1225 + 1325 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
blue then blue - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 410 × 39 = 1290 = 215
blue then blue: 610 × 59 = 3090 = 13 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.215 + 13 = 215 + 515 = 715
Check. The routes not used add up to 815, and 715 + 815 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.rain then dry
dry then rain - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.rain then dry: 0.3 × 0.7 = 0.21
dry then rain: 0.7 × 0.3 = 0.21 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.0.21 + 0.21 = 0.42
Check. The routes not used add up to 0.58, and 0.42 + 0.58 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Use P(event) = 1 − P(opposite event). Find the routes of the opposite event on the tree, multiply along them, add them, and subtract from 1.
- Think about the opposite event. It is easier to find the probability that the event does not happen, then subtract from 1. The routes that do not give "at least one is blue" are:red then red
- Multiply along each of these routes. Each route is "first thing and second thing", so we multiply the branch probabilities.red then red: 58 × 47 = 2056 = 514
- Subtract from 1. The event and its opposite together make probability 1.1 − 514 = 914
Check. Directly: the routes that give the event add up to 914, which agrees. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Watch out
With replacement the second-stage probabilities stay the same. Without replacement they change: the denominator goes down by one. Also check that your final route probabilities add up to 1, and do not add along a branch or multiply between different routes.Exercise 23G Tree diagrams: independent events and replacement
Non-calculatorKey words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 35 × 35 = 925
Check. The routes not used add up to 1625, and 925 + 1625 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.blue then blue
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.blue then blue: 25 × 25 = 425
Check. The routes not used add up to 2125, and 425 + 2125 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then blue
blue then red - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then blue: 35 × 25 = 625
blue then red: 25 × 35 = 625 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.625 + 625 = 1225
Check. The routes not used add up to 1325, and 1225 + 1325 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Use P(event) = 1 − P(opposite event). Find the routes of the opposite event on the tree, multiply along them, add them, and subtract from 1.
- Think about the opposite event. It is easier to find the probability that the event does not happen, then subtract from 1. The routes that do not give "at least one red" are:blue then blue
- Multiply along each of these routes. Each route is "first thing and second thing", so we multiply the branch probabilities.blue then blue: 25 × 25 = 425
- Subtract from 1. The event and its opposite together make probability 1.1 − 425 = 2125
Check. Directly: the routes that give the event add up to 2125, which agrees. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Method. The branches leaving any point add up to 1, and the second spin has the same probabilities as the first.
- Look at the first spin. P(red) = 2/5 and P(blue) = 3/5, and they add to 1.25 + 35 = 1
- Use independence. The second spin is not affected by the first, so after every first result the second-stage branches have the same probabilities as the first spin.Red 2/5 and blue 3/5 at every second stage
- Fill in the ?. After a red first spin the two second branches are red and blue.The first ? is 2/5 (red) and the second ? is 3/5 (blue).
Check. Each pair of second branches adds to 1: 2/5 + 3/5 = 1. ✓
Watch out. Branches leaving one point must always total 1.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 510 × 510 = 25100 = 14
Check. The routes not used add up to 34, and 14 + 34 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
green then green
blue then blue - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 510 × 510 = 25100 = 14
green then green: 310 × 310 = 9100
blue then blue: 210 × 210 = 4100 = 125 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.14 + 9100 + 125 = 25100 + 9100 + 4100 = 38100 = 1950
Check. The routes not used add up to 3150, and 1950 + 3150 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.green then blue
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.green then blue: 310 × 210 = 6100 = 350
Check. The routes not used add up to 4750, and 350 + 4750 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.green then green
green then blue
blue then green
blue then blue - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.green then green: 310 × 310 = 9100
green then blue: 310 × 210 = 6100 = 350
blue then green: 210 × 310 = 6100 = 350
blue then blue: 210 × 210 = 4100 = 125 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.9100 + 350 + 350 + 125 = 9100 + 6100 + 6100 + 4100 = 25100 = 14
Check. The routes not used add up to 34, and 14 + 34 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.hit then hit
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.hit then hit: 14 × 14 = 116
Check. The routes not used add up to 1516, and 116 + 1516 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.hit then miss
miss then hit - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.hit then miss: 14 × 34 = 316
miss then hit: 34 × 14 = 316 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.316 + 316 = 616 = 38
Check. The routes not used add up to 58, and 38 + 58 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Use P(event) = 1 − P(opposite event). Find the routes of the opposite event on the tree, multiply along them, add them, and subtract from 1.
- Think about the opposite event. It is easier to find the probability that the event does not happen, then subtract from 1. The routes that do not give "shaded at least once" are:miss then miss
- Multiply along each of these routes. Each route is "first thing and second thing", so we multiply the branch probabilities.miss then miss: 34 × 34 = 916
- Subtract from 1. The event and its opposite together make probability 1.1 − 916 = 716
Check. Directly: the routes that give the event add up to 716, which agrees. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.rain then rain
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.rain then rain: 0.4 × 0.4 = 0.16
Check. The routes not used add up to 0.84, and 0.16 + 0.84 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.dry then dry
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.dry then dry: 0.6 × 0.6 = 0.36
Check. The routes not used add up to 0.64, and 0.36 + 0.64 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.rain then dry
dry then rain - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.rain then dry: 0.4 × 0.6 = 0.24
dry then rain: 0.6 × 0.4 = 0.24 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.0.24 + 0.24 = 0.48
Check. The routes not used add up to 0.52, and 0.48 + 0.52 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.score then score
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.score then score: 0.7 × 0.7 = 0.49
Check. The routes not used add up to 0.51, and 0.49 + 0.51 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.score then miss
miss then score - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.score then miss: 0.7 × 0.3 = 0.21
miss then score: 0.3 × 0.7 = 0.21 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.0.21 + 0.21 = 0.42
Check. The routes not used add up to 0.58, and 0.42 + 0.58 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Use P(event) = 1 − P(opposite event). Find the routes of the opposite event on the tree, multiply along them, add them, and subtract from 1.
- Think about the opposite event. It is easier to find the probability that the event does not happen, then subtract from 1. The routes that do not give "scores at least one" are:miss then miss
- Multiply along each of these routes. Each route is "first thing and second thing", so we multiply the branch probabilities.miss then miss: 0.3 × 0.3 = 0.09
- Subtract from 1. The event and its opposite together make probability 1.1 − 0.09 = 0.91
Check. Directly: the routes that give the event add up to 0.91, which agrees. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.pass then pass
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.pass then pass: 0.85 × 0.9 = 0.765
Check. The routes not used add up to 0.235, and 0.765 + 0.235 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.pass then fail
fail then pass - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.pass then fail: 0.85 × 0.1 = 0.085
fail then pass: 0.15 × 0.9 = 0.135 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.0.085 + 0.135 = 0.22
Check. The routes not used add up to 0.78, and 0.22 + 0.78 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.fail then fail
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.fail then fail: 0.15 × 0.1 = 0.015
Check. The routes not used add up to 0.985, and 0.015 + 0.985 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.tails then tails
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.tails then tails: 12 × 12 = 14
Check. The routes not used add up to 34, and 14 + 34 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.heads then tails
tails then heads - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.heads then tails: 0.6 × 0.4 = 0.24
tails then heads: 0.4 × 0.6 = 0.24 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.0.24 + 0.24 = 0.48
Check. The routes not used add up to 0.52, and 0.48 + 0.52 = 1. ✓
Watch out. In a “show that” question write the final line clearly: exactly one head has probability 0.48.
Method. Think about what the branches from one point represent.
- Look at one point in the tree. The branches leaving it show every possible result of the next event.e.g. red or blue
- Use what you know about probabilities. Exactly one of these results must happen, and they cannot happen together, so their probabilities add up to 1.sum of probabilities = 1
- Give an example. In the tree with replacement: P(red) + P(blue) = 3/5 + 2/5 = 1.35 + 25 = 1
Check. The same reasoning shows that the probabilities of all complete routes add to 1. ✓
Watch out. A branch probability always uses the same “out of” as the other branches from that point.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then blue
blue then red - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then blue: 410 × 610 = 24100 = 625
blue then red: 610 × 410 = 24100 = 625 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.625 + 625 = 1225
Check. The routes not used add up to 1325, and 1225 + 1325 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
blue then blue
green then green - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 24 × 24 = 416 = 14
blue then blue: 14 × 14 = 116
green then green: 14 × 14 = 116 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.14 + 116 + 116 = 416 + 116 + 116 = 616 = 38
Check. The routes not used add up to 58, and 38 + 58 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Exercise 23H Tree diagrams: dependent events and no replacement
Non-calculatorKey words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 410 × 39 = 1290 = 215
Check. The routes not used add up to 1315, and 215 + 1315 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.blue then blue
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.blue then blue: 610 × 59 = 3090 = 13
Check. The routes not used add up to 23, and 13 + 23 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
blue then blue - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 410 × 39 = 1290 = 215
blue then blue: 610 × 59 = 3090 = 13 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.215 + 13 = 215 + 515 = 715
Check. The routes not used add up to 815, and 715 + 815 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then blue
blue then red - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then blue: 410 × 69 = 2490 = 415
blue then red: 610 × 49 = 2490 = 415 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.415 + 415 = 815
Check. The routes not used add up to 715, and 815 + 715 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Use P(event) = 1 − P(opposite event). Find the routes of the opposite event on the tree, multiply along them, add them, and subtract from 1.
- Think about the opposite event. It is easier to find the probability that the event does not happen, then subtract from 1. The routes that do not give "at least one red" are:blue then blue
- Multiply along each of these routes. Each route is "first thing and second thing", so we multiply the branch probabilities.blue then blue: 610 × 59 = 3090 = 13
- Subtract from 1. The event and its opposite together make probability 1.1 − 13 = 23
Check. Directly: the routes that give the event add up to 23, which agrees. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Method. After the first counter is taken, 4 counters are left. Work out how many of each colour remain.
- If the first counter is red. Red: 3 − 1 = 2 left. Blue: still 2. Total 4.red 2/4, blue 2/4
- If the first counter is blue. Red: 3 left. Blue: 2 − 1 = 1 left. Total 4.red 3/4, blue 1/4
- Find P(both blue). Multiply along the blue then blue route.25 × 14 = 220 = 110
Check. All the routes should add to 1: 6/20 + 6/20 + 6/20 + 2/20 = 1. ✓
Watch out. Without replacement the denominator drops from 5 to 4.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 510 × 49 = 2090 = 29
Check. The routes not used add up to 79, and 29 + 79 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.green then yellow
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.green then yellow: 310 × 29 = 690 = 115
Check. The routes not used add up to 1415, and 115 + 1415 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.red then red
green then green
yellow then yellow - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.red then red: 510 × 49 = 2090 = 29
green then green: 310 × 29 = 690 = 115
yellow then yellow: 210 × 19 = 290 = 145 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.29 + 115 + 145 = 1045 + 345 + 145 = 1445
Check. The routes not used add up to 3145, and 1445 + 3145 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.green then green
green then yellow
yellow then green
yellow then yellow - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.green then green: 310 × 29 = 690 = 115
green then yellow: 310 × 29 = 690 = 115
yellow then green: 210 × 39 = 690 = 115
yellow then yellow: 210 × 19 = 290 = 145 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.115 + 115 + 115 + 145 = 345 + 345 + 345 + 145 = 1045 = 29
Check. The routes not used add up to 79, and 29 + 79 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.dark then dark
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.dark then dark: 38 × 27 = 656 = 328
Check. The routes not used add up to 2528, and 328 + 2528 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.milk then dark
dark then milk - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.milk then dark: 58 × 37 = 1556
dark then milk: 38 × 57 = 1556 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.1556 + 1556 = 3056 = 1528
Check. The routes not used add up to 1328, and 1528 + 1328 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Use P(event) = 1 − P(opposite event). Find the routes of the opposite event on the tree, multiply along them, add them, and subtract from 1.
- Think about the opposite event. It is easier to find the probability that the event does not happen, then subtract from 1. The routes that do not give "at least one dark" are:milk then milk
- Multiply along each of these routes. Each route is "first thing and second thing", so we multiply the branch probabilities.milk then milk: 58 × 47 = 2056 = 514
- Subtract from 1. The event and its opposite together make probability 1.1 − 514 = 914
Check. Directly: the routes that give the event add up to 914, which agrees. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Draw the tree diagram. Write each probability on its branch. The branches from one point must add up to 1. The ends of the branches show every possible outcome.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.black then black
white then white - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.black then black: 610 × 59 = 3090 = 13
white then white: 410 × 39 = 1290 = 215 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.13 + 215 = 515 + 215 = 715
Check. The routes not used add up to 815, and 715 + 815 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.boy then boy
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.boy then boy: 512 × 411 = 20132 = 533
Check. The routes not used add up to 2833, and 533 + 2833 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.girl then boy
boy then girl - Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.girl then boy: 712 × 511 = 35132
boy then girl: 512 × 711 = 35132 - Add the routes. The routes are different ways for the event to happen (this or that), and they cannot happen together, so we add.35132 + 35132 = 70132 = 3566
Check. The routes not used add up to 3166, and 3566 + 3166 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Find the routes through the tree that give the event, multiply the probabilities along each route, then add the routes together.
- Find the routes through the tree that give the event. Each complete route is one possible outcome. Pick out the ones that match.Jo then Jo
- Multiply along each of these routes. Each route means “first thing and then second thing”, so we multiply the branch probabilities.Jo then Jo: 520 × 419 = 20380 = 119
Check. The routes not used add up to 1819, and 119 + 1819 = 1. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Key words. A tree diagram shows every outcome of two or more events in order. The probabilities on the branches from one point always add to 1. Multiply along a route (“and”) and add the routes you want (“or”).
Method. Use P(event) = 1 − P(opposite event). Find the routes of the opposite event on the tree, multiply along them, add them, and subtract from 1.
- Think about the opposite event. It is easier to find the probability that the event does not happen, then subtract from 1. The routes that do not give "Jo wins at least one" are:other then other
- Multiply along each of these routes. Each route is "first thing and second thing", so we multiply the branch probabilities.other then other: 1520 × 1419 = 210380 = 2138
- Subtract from 1. The event and its opposite together make probability 1.1 − 2138 = 1738
Check. Directly: the routes that give the event add up to 1738, which agrees. ✓
Watch out. Multiply along the branches (and), then add between the routes that give the event (or). Do not add along a route.
Method. Without replacement the second probability changes because there is one counter fewer.
- Find what Sam assumed. He used 3/5 twice, which is right only if the first counter is put back.Sam: with replacement
- Find the second probability. After taking one red counter, 2 red counters remain out of 4 counters.P(second red, if the first was red) = 24
- Multiply. Both red means first red and second red.35 × 24 = 620 = 310
Check. Sam’s answer 9/25 = 0.36 is too big; the correct 0.3 is a bit smaller. ✓
Watch out. Without replacement: bottom number drops by 1, and the top number of the colour just taken drops by 1.
Method. Work out each route and compare.
- Find P(red then blue). First red out of 10, then blue out of 9.410 × 69 = 2490 = 415
- Find P(blue then red). First blue out of 10, then red out of 9.610 × 49 = 2490 = 415
- Compare. Both routes use the same numbers 4, 6, 10 and 9, just multiplied in a different order.Both are 4/15, so they are equal.
Check. The final answer for one of each colour is 4/15 + 4/15 = 8/15. ✓
Watch out. Multiplication can be done in any order, so the order of the picks does not change the probability.