When outcomes are not equally likely, you estimate a probability by experiment. Once you know a probability, you can predict how often something should happen.
Relative frequency \(= \dfrac{\text{number of times it happened}}{\text{number of trials}}\), an estimate of the probability that gets more reliable with more trials. Expected frequency \(= \text{probability} \times \text{number of trials}\).
Fig. 1: relative frequency of heads after 10, 20, 50, 100, 200 and 500 throws of a coin (7, 12, 28, 46, 103 and 248 heads); it settles towards 0.5 as the number of trials grows
Section 01
Relative frequency.
In the lab, pick an outcome to count and add trials 1, 10, 100 or 1000 at a time. Watch the relative frequency jump about at first, then settle down close to the true probability as the trials pile up (Fig. 1).
Worked example A drawing pin
A drawing pin is dropped 80 times. It lands point up 37 times. Estimate the probability that it lands point up.
The two ways of landing are not equally likely, so we must experiment.
Relative frequency \(= \dfrac{37}{80} = 0.4625\).
Answer: 0.4625 (about 0.463)
Worked example Is the spinner fair?
A four-sided spinner is spun 200 times. It lands on 1 46 times, on 2 52 times, on 3 61 times and on 4 41 times. Find the relative frequency of landing on 3 and comment on whether the spinner is fair.
Relative frequency of 3 \(= \dfrac{61}{200} = 0.305\).
A fair spinner gives each number probability \(\tfrac{1}{4} = 0.25\), about 50 times in 200 spins.
0.305 is well above 0.25, and 4 is well below, so the spinner may be biased. More spins would make the evidence stronger.
Answer: 0.305; the spinner may be biased towards 3
Examiner trap
A relative frequency is an ESTIMATE. With only 10 throws (Fig. 1) a fair coin gave 0.7. Never decide a dice is biased from a handful of trials.
Section 02
Fair, biased and random.
These three words come up in almost every question on experiments. Know them exactly.
Fair: every outcome is equally likely (a fair dice gives each score probability \(\tfrac{1}{6}\)).
Biased: the outcomes are not equally likely.
At random: every item has the same chance of being chosen; no choosing on purpose.
Reliability: the more trials, the better the estimate. When results are combined, add the frequencies and add the trials.
Worked example Combining results
Ana throws a dice 20 times and gets 8 sixes. Ben throws the same dice 300 times and gets 52 sixes. (a) Whose results give the better estimate? (b) Find the best estimate of P(six). (c) Is the dice fair?
Do not average two relative frequencies: \(\tfrac{1}{2}(0.4 + 0.275)\) gives the small group as much weight as the big one. Add the counts and the trials instead.
Stage 1 · Learn
Check your understanding
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A coin is thrown 50 times and lands on heads 18 times. What is the relative frequency of heads?
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Four students estimate the same probability by experiment. Whose estimate is most reliable?
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A fair dice is rolled 120 times. How many 4s would you expect?
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What does it mean to say a dice is biased?
Section 03
Expected frequency.
In the lab, set a probability and a number of trials, see the expected number, then press Try it once for real. The real count is usually close to the expected one, but rarely exactly equal.
Worked example A fair dice
A fair dice is rolled 300 times. How many sixes would you expect?
\(\text{P(six)} = \tfrac{1}{6}\).
Expected number \(= \tfrac{1}{6} \times 300 = 50\).
Answer: 50
Worked example From a probability
The probability that a certain train is late is 0.15. Over 40 days, how many times would you expect it to be late?
Expected \(= 0.15 \times 40 = 6\).
Answer: 6 days
Worked example From a relative frequency
Using the spinner results from the first section (61 threes in 200 spins), estimate how many times it will land on 3 in 1000 spins.
Best estimate of P(3) \(= 0.305\).
Expected \(= 0.305 \times 1000 = 305\).
Answer: 305
Your turn
P(red) = 0.24 for a biased spinner. It is spun 75 times. How many reds do you expect?
\(0.24 \times 75 = 18\).
Examiner trap
Expected frequency is probability TIMES trials, not trials divided by the probability. \(240 \div 0.35 = 686\) is bigger than the number of spins, which is impossible.
Stage 2 · Exam
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For a biased spinner, P(green) = 0.35. It is spun 240 times. How many times would you expect it to land on green?
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A dice is thrown 250 times and lands on six 61 times. Use these results to estimate the number of sixes in 1000 throws.
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A spinner is spun 100 times: red 23, blue 31, yellow 18 and the rest green. Estimate the number of greens in 450 spins.
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Group A gets 14 successes in 40 trials; group B gets 50 successes in 160 trials. What is the best estimate of the probability of success?
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