GCSE Maths · Probability · Higher · Lesson 23.3

GCSE Maths: Relative frequency, expected outcomes and fairness (Higher)

Part of the GCSE Maths probability chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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23.3

Relative frequency, expected outcomes and fairness

Edexcel 1MA1: P1, P2, P3, P5
Grade 2-6

Sometimes we cannot calculate a probability because the outcomes are not equally likely. Think of a drawing pin: it can land point up or point down, but there is no reason for each to be 1/2. Then we find out by trying it. Each go is a trial, and the number of times the event happens is its frequency. A table of these results is a frequency table. The relative frequency is the fraction of the trials in which the event happened:

relative frequency = frequency ÷ total number of trials

If a pin lands point up 18 times in 50 trials, the relative frequency of “point up” is 18/50 = 9/25 = 0.36. We use the relative frequency as our estimate of the probability.

More trials give better estimates. A small number of trials can be misleading. The graph below shows the relative frequency of heads when a coin was tossed 500 times (the results come from a computer simulation of a fair coin). After 10 tosses the relative frequency was 0.3. After 100 tosses it was 0.55, and after 500 tosses it was 0.478. As the number of trials grows, the relative frequency settles down towards the theoretical probability 0.5. This is why scientists, insurers and doctors collect as much data as they can. But a fair coin will not usually give exactly half heads: there is always some random variation.

Relative frequency of heads (blue) settles near the theoretical probability 0.5 (dashed line) as the number of tosses increases.

Expected outcomes. If we know the probability of an event, we can predict roughly how many times it will happen in a number of trials: expected number = probability × number of trials. If P(6) = 1/6 and a die is rolled 120 times, we expect 1/6 × 120 = 20 sixes. We do not promise exactly 20: it is an average over many sets of trials.

Fair or biased? A fair die has all six outcomes equally likely. To test a die, roll it many times and compare each relative frequency with 1/6. If they are all close, there is no evidence of bias. If one is far away, and the number of trials is large, then the die may be biased. We say “may be” because we can never be completely sure from experiment alone. Throughout this chapter we assume experiments are random (nothing is rigged) and that each trial does not affect the next.

Key ideas

Relative frequency = frequency ÷ total number of trials. It estimates the probability.
More trials usually give a better estimate. Results from few trials can be very different from the true probability.
Expected number = probability × number of trials. It is an average, not a promise.
Compare relative frequencies with the theoretical probability to judge fairness. A fair die does not give exactly equal results.
Relative frequencies from all the possible outcomes of one experiment add up to 1.

Worked example 23.3.1

A spinner is spun 80 times. It lands on red 28 times. Work out the relative frequency of red, in its simplest form.

Key words. Frequency is how many times something happened. Relative frequency = frequency ÷ total number of trials. It is an estimate of the probability. An expected number is probability × number of trials.

Method. Relative frequency is the number of times the event happened divided by the number of trials.

Write the rule. Relative frequency = frequency ÷ total number of trials. It estimates the probability.
relative frequency = frequencytrials
Put the numbers in. The event happened 28 times in 80 trials.
2880
Simplify the fraction. Divide the top and the bottom by their highest common factor, 4. The value does not change but the numbers are smaller.
28 ÷ 4 = 7 and 80 ÷ 4 = 20
2880 = 720

Check. Multiply back: 720 × 80 = 28, which is the frequency 28. ✓

Watch out. The bottom of the fraction is the total number of trials, not the number of times the event did not happen.

Worked example 23.3.2

A fair die is rolled 120 times. How many times would you expect it to land on a 5?

Method. The expected number of times something happens is the probability multiplied by the number of trials.

Write down the rule. We multiply how likely the event is on one trial by the number of trials.
expected number = probability × number of trials
Write down the probability and the number of trials. We need both numbers.
P(5) = 16, number of trials = 120
Find the expected number. Expected number = probability × number of trials. Take 16 of 120: divide by 6, then multiply by 1.
120 ÷ 6 = 20
20 × 1 = 20

Check. Work backwards: 20 ÷ 120 = 0.17, which is the probability we started with. ✓

Watch out. An expected number is an average over many trials. It does not promise exactly that many will happen.

Worked example 23.3.3

A factory checks 500 bulbs. 12 are faulty. (a) Estimate the probability that a bulb is faulty. (b) The factory makes 8000 bulbs in a week. Estimate how many are faulty.

Method. Use the relative frequency of faulty bulbs as the probability, then multiply by the number of bulbs.

Work out the relative frequency. Faulty bulbs divided by the number checked.
relative frequency = 12500 = 3125
Use it as the estimated probability. We use the experiment to estimate the probability.
P(faulty) ≈ 3125 = 0.024
Multiply by the number of bulbs made. Expected number = probability × number of trials.
3125 × 8000 = 8000 ÷ 125 × 3 = 64 × 3 = 192

Check. 12 faulty in 500 means 24 faulty in 1000, so in 8000 we expect 24 × 8 = 192. ✓

Watch out. The answer is an estimate, so we write “about 192” and not “exactly 192”.

Worked example 23.3.4

Dan tosses a coin 200 times and gets 124 heads. Is there evidence that the coin is biased?

Method. Work out the relative frequency of heads and compare it with 1/2, which is the probability for a fair coin.

Work out the relative frequency of heads. Frequency of heads divided by the number of tosses.
124200 = 3150 = 0.62
Find the expected number of heads for a fair coin. A fair coin gives P(heads) = 1/2.
12 × 200 = 100 heads
Compare. 124 heads is quite a lot more than 100. With 200 trials this is a fairly big difference.
0.62 compared with 0.5
Write a conclusion. Because the gap is large for so many tosses, it suggests the coin may be biased towards heads. We cannot be certain.
There is some evidence the coin is biased.

Check. If the coin were fair, 100 heads is the average. 124 is 24 more, which is unusual but not impossible. More tosses would give a stronger conclusion. ✓

Watch out. Never say a coin is “definitely” biased from one experiment.

Watch out

A relative frequency from a few trials can be far from the real probability. Tom tossing a coin 10 times and getting 7 heads does not prove that P(heads) = 0.7. Also, the expected number is an average: expecting 20 sixes in 120 rolls does not mean you will get exactly 20.

Exercise 23C Relative frequency

Non-calculator

FWrite down the relative frequency of the event. Give each answer as a fraction in its simplest form.

(a)The event happened 9 times in 30 trials

(b)The event happened 24 times in 80 trials

(c)The event happened 15 times in 60 trials

(d)The event happened 35 times in 140 trials

Done

FWork out the relative frequency of each event as a decimal.

(a)6 out of 20 trials

(b)27 out of 60 trials

(d)7 out of 8 trials

Done

FTom spins a spinner 80 times. The table shows his results. Find the relative frequency of each colour, as a fraction in its simplest form.

Colour	Red	Blue	Green
Frequency	24	31	25

(a)Red

(b)Blue

(c)Green

(d)Add the three relative frequencies

Done

FA coin is tossed several times. The table shows the number of heads after different numbers of tosses. Work out the relative frequency of heads each time, as a decimal.

Number of tosses	20	50	100	200
Number of heads	12	27	52	98

(a)After 20 tosses

(b)After 50 tosses

(c)After 100 tosses

(d)After 200 tosses

(e)Which result is the best estimate of P(heads)? Explain.

Done

PSIn a survey 200 students were asked how they travel to school. 38 walk, 62 take the bus, 70 come by car and the rest cycle. A student is chosen at random from the 200. Estimate the probability that the student...

(a)walks

(b)cycles

Done

RTom tosses a coin 10 times and gets 7 heads. He says: “P(heads) must be 0.7.” Explain why Tom may be wrong.

Done

RElla spun a spinner 20 times and landed on red 6 times. Raj spun the same spinner 200 times and landed on red 58 times. Who has the better estimate of P(red)? Give a reason, and write both estimates as decimals.

Done

RA fair coin was tossed 400 times by a computer. The table gives the number of heads after 20, 50, 100, 200 and 400 tosses and the graph shows the relative frequency of heads.

Number of tosses	20	50	100	200	400
Heads	12	28	50	102	184

(a)Work out the relative frequency of heads after 50 tosses, as a decimal.

(b)Work out the relative frequency of heads after 400 tosses, as a decimal.

(c)The theoretical probability of heads is 0.5. Which is closer to 0.5: the result after 20 tosses or after 400 tosses?

Done

PSA bag contains 20 counters that are red or blue. Sam picks a counter, records its colour and puts it back. He does this 150 times and gets red 93 times. Use his results to estimate the number of red counters in the bag.

Done

Exercise 23D Expected outcomes, fair and biased

Calculator allowed

FA fair six-sided die is rolled. How many times would you expect to roll...

(a)a 3 in 60 rolls

(b)an even number in 120 rolls

(c)a number bigger than 4 in 300 rolls

(d)a 6 in 450 rolls

Done

FWork out the expected number of times that each event happens.

(a)A fair coin is tossed 500 times: heads

(b)A spinner has P(red) = 3/8 and is spun 200 times: red

(c)The probability of rain on a day is 0.2: rainy days in 365 days

(d)P(a part is faulty) = 2%: faulty parts in 5000

Done

PSA drawing pin is dropped 60 times and lands point up 24 times. Use his results to answer the questions.

(a)Relative frequency of point up

(b)Estimate for 150 drops

Done

PSAt a fair, a game costs £2 to play. The probability of winning is 112 and a win pays £10 prize money. In one day the game is played 300 times. Work out the expected profit for the fair.

Done

PSMaria rolls a die 120 times. The table shows her results.

Score	1	2	3	4	5	6
Frequency	17	23	19	25	16	20

(a)How many times would you expect each score if the die is fair?

(b)Which score is furthest from the expected number?

(c)Is there evidence that the die is biased? Give a reason.

Done

RMaya tosses a fair coin and gets heads 5 times in a row. She says: “Tails is more likely next time, to even things out.” Is Maya correct? Explain.

Done

RDie A is rolled 30 times and shows a 6 on 5 rolls. Die B is rolled 300 times and shows a 6 on 70 rolls. Which die is more likely to be biased? Show your working.

Done

PSThe probability that Ben wins a game is 0.15. Work out the least number of games Ben must play to expect to win at least 12 games.

Done

RExplain why an expected number of wins does not have to be a whole number, and why it does not tell us exactly how many wins there will be.

Done