Question

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.

Answer 1

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) $= 1-\frac{1}{6}-\frac{1}{3}=\frac{1}{2}$