Probability- Conditional Probability

[Roni1985]

Homework Statement

1 and 2 take turns shooting at a target. Each time 1 shoots he hits the target with probability p₁; 2 hits it with probability p₂ each time. 1 shoots first. They stop when the target has been hit twice. What is the probability that...

a) The first hit was by 1 ?
b) Both hits were by 1?
c)Both hits were by 2?

Homework Equations

Conditional probability?

The Attempt at a Solution

Actually I'm a little (very) rusty. I took probability a year ago, and I just don't know how to start it.
I'm totally lost here.

Would appreciate any help.

Thanks,
Roni.

 

[Office_Shredder]

Could you, for example, find the probability that the first time the target is hit it is by 1 on his second shot? How about if the first time the target is hit is by player 1 and it's his third shot?

 

[Roni1985]

Could you, for example, find the probability that the first time the target is hit it is by 1 on his second shot? How about if the first time the target is hit is by player 1 and it's his third shot?

well,
I think I should follow this:
first shot- p₁

second shot- q₁*q₂*p₁

third shot- q₁*q₂*q₁*q₂*p₁

nth shot- (q₁)^n-1*(q₂)^n-1*p₁

but what's my n?
this is how I leave it ?

Then, I don't need to use conditional probability here?thanks.

 

[Roni1985]

Hello guys,
I'm still kinda lost :\

would appreciate any help ...

 

[Office_Shredder]

You're going to need to use conditional probability. If the first shot was hit by 1:

It was either hit by him on the first shot, or the second shot, or the third shot, or the fourth shot, etc. So the probability that 1 hits the first shot can be written in terms of the probabilities he gets the first shot on his nth shot.

 

[Roni1985]

You're going to need to use conditional probability. If the first shot was hit by 1:

It was either hit by him on the first shot, or the second shot, or the third shot, or the fourth shot, etc. So the probability that 1 hits the first shot can be written in terms of the probabilities he gets the first shot on his nth shot.

Oh, its got to be an infinite geometric sum...
thanks for your help :\

 

[Office_Shredder]

You can do the same type of thing for the other problems. For example, the probability that the first guy gets both hits is based on the probability he gets two hits in his first two shots, and the other guys misses, and the probability that he gets two hits in his first three shots and the other guy misses, etc.