Banachs matchcbox problem-Probability

[kangkan]

Mr. Stephan kept two matchboxes, one in each pocket. Each box
contained exactly n matches. Whenever he wanted a match he reached at
random into one of his pockets. When he found that the box he picked
was empty, what is the probability that the other one has exactly k
matches

I am unable to make sense of the wikipedia solution.

[Arushi :))]

I think sir should help

[Akshay Jain]

let the 2 match boxes be 1 and 2
the choice of 2nd box is regrded as a success and 1st as a failure
mr stephan selects a box at random so P(selectin a match box)=1/2
the 2nd box will b found empty if it is selected for (n+1)th time and at the same time the other box shud contain k matchsticks, so n-k sticks are already drawn from dat box.
so the n-k failures (2nd box is chosen) shud precede the (n+1)th success
the total no. of trials are n+n+1-k(total matchsticks drawn and at last 1st box is found empty vd k sticks in 2nd) and there shub b n-k failures and n successes
applying negetive bionomial distribution
requared probablity is (2n-kCn)*(1/2)^n*(1/2)^n-k*1/2 (P1)
similar calculation when 1st is found with k sticks and 2nd is found empty(P2)
total prob is P1+P2=(2n-kCn)*(1/2)^n*(1/2)^n-k
 

[kangkan]

Thanks akshay :)