Question #210116

A monkey Is to demonstrate that she recoginizes colors by tossing one red, one black,and one white ball into boxes of the same respective colors, one ball to a boOx. If themonkey las not leamed the colorS and nerely tosses one ball into each box al random,find the probabilities of the following results:

a. There are no color matches.

b.There i5 exactly one color match.

Expert's answer

Let "A_1" = Color match in red box

Let "A_2" = Color match in black box

Let "A_3" = Color match in white box

Part 1

P(No color match) = 1- P[ Atleast one match]

P(No color match)"1-P[A_1UA_2UA_3]"

P(No color match)"1-[P(A_1)+P(A_2)+P(A_3)-P(A_1nA_2)-P(A_2nA_3)-P(A_1nA_3)+P(A_1nA_2nA_3)]"

"1-[\\frac{1}{3}+\\frac{1}{3}+\\frac{1}{3}-\\frac{1}{6}-\\frac{1}{6}-\\frac{1}{6}+\\frac{1}{6}]=\\frac{1}{3}"

Part 2

P(exactly one match) = 1- [P(All match)+P(no match)]

P(All match) "= P(A_1)P(A_2\/A_1)P(A_3\/A_1nA_2) = \\frac{1}{3}(\\frac{1}{2})(1)=\\frac{1}{6}"

Now P(Exactly one match) "= 1-\\frac{1}{6}-\\frac{1}{3}=\\frac{1}{2}"

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