Monday, March 9, 2015

Unizor - Probability - Easy Problems 6

Problem A
Assume you have a box filled with N balls of equal sizes and weights, but of n different colors: there are A1 balls of the first color, A2 balls of the second color,..., An balls of the nth color in the box.
So, Σi=1...n (Ai)=N).
You randomly pick M balls from the box.
What is the probability that you pick exactly Bi balls of the ith color (where i=1,2,...,n and Σi=1...n (Bi)=M)?
C[A1,B2]·C[A2,B2]·...·C[An,Bn] / C[N,M].

Problem B
Imagine the following simple game - actually, a kind of a primitive pinball without controls. A player puts a small ball into a spring launcher on a board with many pins and, as the spring released, the ball randomly moves on the board hitting the pins and eventually ends up in one of the holes at the base of a board with certain number of points associated with it.
There is no restriction in the number of balls going through the same hole, they just fall through into some receptacle. Assume that the probability of a ball to fall through any hole is the same for all holes.
There are M different holes on the board and a player plays N times.
Determine the distribution of probabilities of N balls randomly falling through M holes.
Answer: N!/[(M^N)·K1!·K2!·...·KM!)]

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