Expected outcomes and experiments

When the outcomes are not equally likely, estimate the probability from an experiment:

relative frequency = number of times it happened ÷ number of trials
expected number = probability × number of trials

1. a spinner lands on blue 36 times in 120 spins
2. relative frequency = 36 ÷ 120 = 0.3
3. in 500 more spins, expect 0.3 × 500 = 150 blues

An expected number is an estimate, not a promise: the real count will usually be a little different.

More trials give a more reliable estimate. If three people did 30, 50 and 300 throws, the 300 is the best single estimate. Better still, pool everything: add all the successes and all the trials, then divide.

1. results: 9 out of 30, 18 out of 50, 111 out of 300
2. total successes: 9 + 18 + 111 = 138
3. total trials: 30 + 50 + 300 = 380
4. best estimate = 138/380 = 69/190

Do not average the three relative frequencies: that treats 30 throws as if they were worth as much as 300.

On a graph of relative frequency against the number of trials, the line wobbles at first and settles down. The last point uses the most trials, so it is the best estimate.

Work out the number you would expect if it were fair, then compare. A fair dice thrown 600 times should give about 600 ÷ 6 = 100 sixes. 104 sixes is close: small differences happen by chance. 180 sixes is a long way off, so the dice is probably biased. Never say "must": an experiment gives evidence, not proof.

For a game, work out the expected number of winners first, then the money in and the money out.

1. 200 players pay £1 each; P(win) = 1/8; prize £5
2. expected winners = 1/8 × 200 = 25
3. takings = 200 × £1 = £200
4. prizes = 25 × £5 = £125
5. expected profit = £200 − £125 = £75

For two independent events, multiply the probabilities first: P(heads and a six) = 0.4 × 1/6, then multiply by the number of trials.

Spin a biased spinner 20 times, then 500 times, and watch the relative frequency settle down.