Answered: Solve using the additive law of…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Question

Solve using the additive law of probability:

A monkey is to demonstrate that she recognizes colors
by tossing one red, one black, one white ball, one orange ball, and one purple ball into boxes of
the same repsective colors, one ball to a box. If the monkey has
not learned the colors and merely tosses one ball into each box
at random, find the probabilities of the following results:
a. There are no color matches.
b. There is exactly one color match.

Expert Solution

Step 1: There are no color matches.

Understand the given problem :

The problem involves a monkey tossing 5 different colored balls into 5 different colored boxes. The goal is to find the probabilities of two specific outcomes:

a. There are no color matches.

b. There is exactly one color match.

Balls: There are 5 balls of different colors: red, black, white, orange, and purple.
Boxes: There are 5 boxes, each of the same respective colors as the balls.

: Total Permutations:

There are 5 distinct balls to toss into 5 distinct boxes, so the total number of permutations is $5! .$

Number of Derangements:

The number of derangements $(! n)$ represents the number of permutations where no element appears in its original position. In this case, we want to calculate $! 5.$

Calculate Probability:

The probability of no color matches is given by the number of derangements divided by the total permutations.
Probability(no color matches) $equals fraction numerator space factorial 5 over denominator 5 factorial end fraction$
Now, calculate:
- !5 = 44 (number of derangements of 5 elements)
- 5! = 120 (total permutations)
Probability(no color matches) $equals fraction numerator space 44 over denominator space 120 space end fraction space equals space 11 over 30$
So, the probability of no color matches is $11 over 30$
Step 5: Understand The Probability of Exactly One Color Match:

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