Question

here are n bins of which the k-th bin contains k−1 blue balls and n−k red balls. You pick a bin at random and remove two balls at random without replacement. Find the probability that:
• the second ball is red;
• the second ball is red, given that the first is red.

Answer 1

Answer:

Explanation:

etCibe the colour of theith ball.  In each bin, there are a total of (k−1) + (n−k) = (n−1)balls.  Of these half are blue and the other half are red (verify∑nk=1k−1 =∑nk=1n−k).The probability of the second ball being red is equal to the probability of the second ball beingred given that the first ball was either red or blue.For a particular bin we have,P(C2=red) =(n−k)(n−1)(n−k−1)(n−2)+(k−1)(n−1)(n−k)(n−2)=n−kn−1Considering all bins, we haveP(C2=red) =n∑k=1n−kn(n−1)=12The probability of the second ball being red given that the first ball was red,P(C2=red|C1=red) =P(C2=red,C1=red)P(C1=red)Now,P(C1=red) =12.1

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