Question

A group of N girls play the following game in the snow. Each girl removes one...

A group of N girls play the following game in the snow. Each girl removes one of her mittens. The mittens are put in one big pile and then one blindfolded girl redistributes back the mittens randomly, so that each girl is given back one mitten. You can assume that the mittens are distinguishable so that each girl can recognize her own. Note that you can identify the possible outcomes of this experiment with the permutations of the N mittens in the following way. Make the girls form a line and label their mittens with the numbers from 1 to N, starting from the first girl in line. At the beginning of the experiment each girl is holding her own mitten, so that mitten #1 is in the hands of girl #1, mitten #2 is in the hands of girl #2 etc. . . . At the end of the experiment the mittens have been reordered. You can assume that all outcomes are equally likely.

The question you will examine is: What is the probability that at least one girl gets back her own mitten?

Let Ai be the event “girl number i gets back her own mitten” and notice that we are interested in P(A1 ∪ A2 ∪...∪AN).

(a) Use equation P(A∪B)=P(A)+P(B)−P(A∩B)  to find the probability that at least one girl gets back her own mitten when there only two girls (N=2).

(b) Use equation P(A∪B∪C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C ) + P(A∩B∩C) to find the probability that at least one girl gets back her own mitten when there only three girls (N=3).

(c) Write a general formula to find the probability that at least one girl gets back her own mitten in a group of N girls.

Math   Statistics-And-Probability   
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Answer #1

sry,for the untidy upload,it was a huge calculation though. FYI the first picture,P(A)=1/3,in can if you don't understand what's written.

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