Question

Bill and two other people play a game where the first to roll an even number using a standard number cube wins. What's the probability that Bill, by going first, wins?

Answer 1

`~~57.1%`

Explanation:

Bill can win on the first roll if we rolls an even number, and so the probability is:

`1/2`.

However, if he rolls an odd number, the other two people must also roll an odd number for Bill to have another chance. And so for Bill to win on his second roll, there is a probability of:

`1/2xx1/2xx1/2xx1/2=(1/2)^4=1/16`

(the first three rolls are odd and then Bill rolls an even number). And so the probability of Bill winning on either the first or second roll is:

`1/2+1/16=9/16`

We can continue to view this question following this pattern:

`(1/2)^1+(1/2)^4+(1/2)^7+(1/2)^10+...`

`1/2+1/16+1/128+1/1024+...`

and so roughly the probability is

`512/1024+64/1024+8/128+1/1024=585/1024~~57.1%`