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In a series of 5 Test Matches the England cricket captain only won the toss once. |
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I bought 8 pens from a shop and 1 of them did not work. | |
| In a random sample of 100 people, 8 said they would vote for the Green party. |
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Over several years a woman gives birth to 6 children, of which 5 were girls. | |
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Each situation involves an unpredictable event with two possible outcomes. It is traditional to label one outcome as "success" and the other as "failure". The captain may guess correctly (success) or incorrectly; the pen may work (success) or may not work; the person may vote Green (success) or for some other party; the child may be a girl (success) or a boy. In each situation there is a given number of trials of this event. We call this number n. Thus there were n=5 matches, n=8 pens, n=100 voters and n=6 children. Each trial of the event is independent of the others. The fact that the captain guessed wrongly in the first three matches does not make it more (or less) likely that he will guess correctly in the fourth match. Provided the pens were all the same brand, why should the fact that one works have any effect on another? The voters were selected at random and could not therefore influence each other. As several English kings discovered, having several princesses does not make it more likely that the next child will be a prince! Since the trials are independent the probability of each outcome remains constant. We call the probability of a success p and the probability of failure q. There is a 50% chance that the captain wins any toss so p=0.5=q. There is perhaps a 1% chance that any pen does not work so p=0.99 and q=0.01. Maybe 5% of people vote Green so p=0.05 and q=0.95. Approximately 50% of children born are girls so p=0.5=q. Notice that q=1-p. It is important to realise that the number of successes we get in our n trials depends on chance. The fact that 50% of children born are girls does not mean that in every 6 children 3 will be girls. It is possible to get no girls, or all girls, or indeed any number in between. Common-sense tells us that 3 girls is more likely than 5 girls, but how can we calculate the probabilities of getting 0, 1, 2 … 6 girls in 6 births. This is where the Binomial Distribution comes in.
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To continue on the recommended route click on the Derivation button at the top of the page.
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Copyright © 1997 Neville Hunt |
