A dice is rolled 120 times. It lands on six 31 times.
- (a)Work out the relative frequency of rolling a six.[1]
- (b)Does the dice appear to be fair? Give a reason.[1]
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There is no formula for the probability that a drawing pin lands point up. The only way to find out is to drop it many times and count. The fraction of trials with that outcome is the , and it is the best estimate of the probability when theory cannot help.
The big picture
Theoretical probability comes from equally likely outcomes, like the six faces of a fair dice. Experimental probability comes from data. The two are connected: as the number of trials grows, relative frequency settles closer to the true probability.
Once you have a probability — from theory or experiment — you can predict how often something should happen, and compare real results with those expectations to decide whether a dice, spinner or coin is fair.
What you'll learn
Relative frequency .
A drawing pin dropped 300 times lands point up 186 times: relative frequency .
Using the same 400 spins (A: 92, B: 108, C: 64, D: 136), estimate the probability of landing on D.
Expected number .
Using the drawing pin estimate, in 800 drops we expect to land point up.
The probability that an item from a factory is faulty is 0.08. How many faulty items would you expect in a batch of 750?
Tip — An expected value is a prediction, not a promise. Real results vary around it.
A fair dice should show each number about of the time. Rolling it 60 times, you expect each number about 10 times.
Getting a six 18 times out of 60 (relative frequency 0.3) suggests bias, but 60 rolls is not many. More trials would give more reliable evidence.
Combining results from several groups gives a larger total and a better estimate.
Small samples jump around. Ten coin flips can easily give 7 heads, but 7000 heads in 10 000 flips would be strong evidence of an unfair coin.
Think like an examiner
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Stretch yourself
Asha rolls a dice 50 times and gets a 4 on 13 of them. Ben rolls the same dice 250 times and gets a 4 on 42. (a) Whose results give the better estimate of P(4)? (b) Combine the results to estimate P(4). (c) Does the dice appear to be biased towards 4?
Hint — For (c), compare the combined estimate with .
Questions students ask
Key takeaways
Exam-style questions
2 original questions · 5 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.
A dice is rolled 120 times. It lands on six 31 times.
Ana flips a coin 20 times and gets 6 heads. Ben flips the same coin 500 times and gets 240 heads.
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