John and Steve take turns to throw darts until either of them hits the bullseye...

[a.k.eriksson]

John and Steve take turns to throw darts until either of them hits the bullseye of the dartboard. The probabilities for hitting the bullseye are 0,13 for John and 0,09 for Steve. Find the probability that John hits the bullseye first if John throws first and casts are considered independent events.

I have found that this should be a geometric distribution but Im not sure how to continue to calculate this when there isn't any limit for the number of throws.

 

[pka]

John and Steve take turns to throw darts until either of them hits the bullseye of the dartboard. The probabilities for hitting the bullseye are 0,13 for John and 0,09 for Steve. Find the probability that John hits the bullseye first if John throws first and casts are considered independent events.

I have found that this should be a geometric distribution but Im not sure how to continue to calculate this when there isn't any limit for the number of throws.

If John wins it has to be on an odd throw.
The probability that John wins on the first throw is \(\displaystyle 0.13 \)
The probability that John wins on the third throw is \(\displaystyle (0.87)(0.91)(0.13) \)
The probability that John wins on the fifth throw is \(\displaystyle (0.87)^2(0.91)^2(0.13) \)

Now you need to report back explaining why those work.
Also what the infinite series is that adds up all the odd throws?

 

[a.k.eriksson]

If John wins it has to be on an odd throw.
The probability that John wins on the first throw is \(\displaystyle 0.13 \)
The probability that John wins on the third throw is \(\displaystyle (0.87)(0.91)(0.13) \)
The probability that John wins on the fifth throw is \(\displaystyle (0.87)^2(0.91)^2(0.13) \)

Now you need to report back explaining why those work.
Also what the infinite series is that adds up all the odd throws?

The probability that John hits the bullseye is 0,13 and for a miss it is 1-0,13=0,87.
The probability that Steve hits the bullseye is 0,09 and for a miss it is 1-0,09=0,91.

f(1)=P(John hits the bullseye in the first throw)=0,13
f(2)=P(John miss the bullseye in the first throw and Steve hits the bullseye in the second throw)=(0,87)(0,09)
f(3)=P(John miss the bullseye in the first throw but hits the bullseye in the third throw and Steve miss the bullseye in the second throw)=(0,87)(0,91)(0,13)
f(4)=P(....)=(0,87)^2(0,91)(0,09)
f(5)=P(....)=(0,87)^2(0,91)^2(0,13)

So far I get it, John can only win if it is a odd throw. But now since you cant calculate this to infinity I need to use the geometric sum.
And its here I think things get tricky.

0,13*( (0,87)^0(0,91)^0 + (0,87)^1(0,91)^1 + (0,87)^2(0,91)^2 + (0,87)^3(0,91)^3 and so on......) is this then the same as 0,13*(1/(1-0,87*0,91))=0,62 ??