Another Probability and Odds question

[JellyFish]

The Question asks:

Frank and June play darts. They take turns shooting at a target. The winner is the one who manage to hit the bull's eye center first. Previous contests credited Frank with 2 to 5 odds to hit the center in one shot, while June's odds were 3 to 6. If June shoots first, what is the probability that June beats Frank?

I converted the odds to probability,

P(F) = 2/7 and P(J) = 1/3

To look at them more clearly, P(F) = 6/21 and P(J) = 7/21

So I know that the probability of June winning after just one shot is 7/21 or 1/3. But I'm not sure how to use this to compute the probability than June beats Frank.

Thank you for your help

 

[Deleted member 4993]

The Question asks:

Frank and June play darts. They take turns shooting at a target. The winner is the one who manage to hit the bull's eye center first. Previous contests credited Frank with 2 to 5 odds to hit the center in one shot, while June's odds were 3 to 6. If June shoots first, what is the probability that June beats Frank?

I converted the odds to probability,

P(F) = 2/7 and P(J) = 1/3

To look at them more clearly, P(F) = 6/21 and P(J) = 7/21

So I know that the probability of June winning after just one shot is 7/21 or 1/3. But I'm not sure how to use this to compute the probability than June beats Frank.

Thank you for your help

probability of June winning = 1/3 + (2/3 * 5/7)*1/3 + (2/3 * 5/7)[sup:35op3yk4]2[/sup:35op3yk4]*1/3 + (2/3 * 5/7)[sup:35op3yk4]3[/sup:35op3yk4]*1/3 +(2/3 * 5/7)[sup:35op3yk4]4[/sup:35op3yk4]*1/3 ....

Do you see a pattern? Do you see a geometric series? - sum it...

 

[JellyFish]

I understand that you started with the probability of June hitting in the first shot then added this same probability multiplied by the probabilites against both June and Frank and then each time the exponent grows. I guess I just don't understand why you are using this method or how to know when to stop?

 

[Deleted member 4993]

I understand that you started with the probability of June hitting in the first shot then added this same probability multiplied by the probabilites against both June and Frank and then each time the exponent grows. I guess I just don't understand why you are using this method or how to know when to stop?

It does not stop - it is a convergent geometric series, extends to infinite number of terms - find the sum (it should be a little bit less than 2/3)