A fair spinner has 8 equal parts, only 1 of them red. About how many spins should give red about 37 times?

296 spins. P(red) = 1/8. Trials × 1/8 = 37, so trials = 37 × 8 = 296.

A spinner has only red, yellow and blue. After many spins, the relative frequencies recorded are red 0.43, yellow 0.38 and blue 0.19. At least one of them must be wrong. True or false?

False. All the outcomes' relative frequencies must add to 1: 0.43 + 0.38 + 0.19 = 1.

Tsz-ching tosses a coin many times and records the relative frequency of heads: 0.6 after 10 tosses, 0.54 after 100 and 0.501 after 1000. After 1000 tosses, heads had come up □ times; with the most reliable estimate, 1000 more tosses should give about □ heads. Fill in each blank

501, 501. 1000 × 0.501 = 501. The relative frequency after 1000 tosses is the most reliable: 0.501 × 1000 = 501.

Probability · Ages 14–15

Relative frequency and expected frequency

Relative frequency = frequency ÷ trials; expected frequency = probability × trials.

How to do it

Method

Relative frequency = frequency ÷ trials; expected frequency = probability × trials

46 ÷ 200 = 0.23; 0.4 × 300 = 120

Another way

Against the theory

180 ÷ 600 = 0.3, far more than 16

In full

In an experiment, the number of times an outcome happens is its frequency; frequency ÷ the number of trials is its relative frequency (the experimental probability), which estimates the probability. A die rolled 200 times shows a 6 on 46 of them: the relative frequency of a 6 is 46 ÷ 200 = 0.23. All the outcomes' relative frequencies add to 1. The more trials, the closer the relative frequency usually comes to the true probability, so the better the estimate: 200 rolls tell you more than 20. Expected frequency = probability × number of trials: with a probability of 0.4 over 300 trials, expect about 0.4 × 300 = 120; the probability can come from equally likely outcomes or be estimated from a relative frequency. Is a die biased? One way, compare the counts: a fair die rolled 600 times should show each number about 600 × 16 = 100 times; 180 sixes is far more than 100 over many rolls, so the die is probably biased. Another, compare the relative frequency with the theory: 180 ÷ 600 = 0.3, much more than 16, the same conclusion. But over only 12 rolls, even 4 sixes are too few to conclude anything; and no experiment ever proves a die is fair.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
A fair spinner has 8 equal parts, only 1 of them red. About how many spins should give red about 37 times?
Show the answer and working
Answer: 296 spins
P(red) = 18. Trials × 18 = 37, so trials = 37 × 8 = 296.
Example 2
A spinner has only red, yellow and blue. After many spins, the relative frequencies recorded are red 0.43, yellow 0.38 and blue 0.19. At least one of them must be wrong.
True or false?
Show the answer and working
Answer: False
All the outcomes' relative frequencies must add to 1: 0.43 + 0.38 + 0.19 = 1.
Example 3
Tsz-ching tosses a coin many times and records the relative frequency of heads: 0.6 after 10 tosses, 0.54 after 100 and 0.501 after 1000. After 1000 tosses, heads had come up times; with the most reliable estimate, 1000 more tosses should give about heads.
Fill in each blank
Show the answer and working
Answer: 501, 501
1000 × 0.501 = 501. The relative frequency after 1000 tosses is the most reliable: 0.501 × 1000 = 501.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Four people each toss the same kind of drawing pin a different number of times and record the relative frequency of point up. Whose result gives the most reliable estimate of the probability of point up?
- A. The one who tossed it 120 times
- B. The one who tossed it 45 times
- C. The one who tossed it 1000 times
- D. The one who tossed it 6 times
Choose the right answer
Answer:
2.A spinner's pointer stops on red with probability 0.55. Spun 540 times, it should stop on red about times.
Answer:
3.A fair spinner has 8 equal parts, only 1 of them red. About how many spins should give red about 14 times?
Answer:
4.A spinner has only red, yellow and blue. After many spins, the relative frequencies recorded are red 0.48, yellow 0.31 and blue 0.2. At least one of them must be wrong.
True or false?
Answer:
5.A fair spinner has 10 equal parts, 3 of them red. Spun 1800 times, it stops on red 531 times. How far is the actual count from the expected one?
Answer:

All the answers

1. C. The one who tossed it 1000 times
2. 297
3. 112 spins
4. True
5. 9 times

Free worksheet: Relative frequency and expected frequencyWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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