Math Is Fun Forum

A machine fires its first shot and hits the target, but it misses its second shot. All its subsequent shots have a probability of hitting the target equal to the proportion of targets hit beforehand. For example, if it hits 5 out of the first 8 shots, then the 9th shot has a probability of 5/8 to hit the target. What is the probability that it hits exactly 50 out of its first 100 shots. Leave your answer in exact form (e.g. a/b).

If you're not good at word problems, it can be written like this
X~((x+1)/(y+2),98) (x is number of successful trials, y is number of trials at that point)
P(X=49)=A
A=a/b
Find a/b

The work i've done on it is above and i know i can work this problem like a tree diagram, but no one wants to be drawing 2^100 branches come the last trial!
I have worked out that the last trial must be at 49 or 50 successes though, both of which end in 49/99 probability of ending on exactly 50 successes, but i can't reverse engineer beyond that.

I have just started A-levels and came across this problem on a sharing site and I would like a easier/faster method as well as an answer if anybody can think of one. Thank you for your time.