On the average, how many times must a die be thrown until one gets a 6?
Solution:
Let be the probability of a 6 on a given trial. Then the probabilities of success for the first time on each trial are (let ):
Trial | Probability of success on trial |
---|---|
1 | |
2 | |
3 | |
. | . |
. | . |
. | . |
The sum of the probabilities is
The mean number of trials, , is by definition,
Note that our usual trick for summing a geometric series works:
Subtracting the second expression from the first gives
or
Consequently,
and
We see that , and so .
On the average, a die must be thrown 6 times until one gets a 6.
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