Understandable Statistics: Concepts and Methods
Understandable Statistics: Concepts and Methods
12th Edition
ISBN: 9781337119917
Author: Charles Henry Brase, Corrinne Pellillo Brase
Publisher: Cengage Learning
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Chapter 5.2, Problem 17P

a.

To determine

Compute the probability that at least 6 wolves were males, at least 6 wolves were females, and less than 4 wolves were females before 1918.

a.

Expert Solution
Check Mark

Answer to Problem 17P

The probability that at least 6 wolves were males is 0.740.

The probability that at least 6 wolves were females is 0.473.

The probability that less than 4 wolves were females is 0.135.

Explanation of Solution

Calculation:

A trial determines the sex of a wolf with two possible outcomes, success or failure. Consider success as male and failure as female.

The total number of wolves (or trials) is n=12.

The probability of male wolves (or success) is p=0.55.

The probability of female wolves (or failure) is q=0.45.

Binomial probability:

The probability of r successes out of n trials is given below:

P(r)=Cn,rprqnr

Here, n is the number of trials, r is the number of successes, p is the probability of success, and q is the probability of failure.

Probability that at least 6 wolves were males:

The probability that at least 6 wolves were males is calculated as given below:

P(r6)=P(6)+P(7)+P(8)+P(9)+P(10)+P(11)+P(12)=C12,6(0.55)6(0.45)6+C12,7(0.55)7(0.45)5+C12,8(0.55)8(0.45)4+C12,9(0.55)9(0.45)3+C12,10(0.55)10(0.45)2+C12,11(0.55)11(0.45)1+C12,12(0.55)12(0.45)0=0.212+0.223+0.170+0.092+0.034+0.008+0.001=0.740

Therefore, the probability that at least 6 wolves were males is 0.740.

Probability that at least 6 wolves were females:

Six or more females are same as less than or equal to males.

The probability that at least 6 wolves were females is calculated as given below:

P(r6)=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=C12,0(0.55)0(0.45)12+C12,1(0.55)1(0.45)11+C12,2(0.55)2(0.45)10+C12,3(0.55)3(0.45)9+C12,4(0.55)4(0.45)8+C12,5(0.55)5(0.45)7+C12,6(0.55)6(0.45)6=0.000+0.001+0.007+0.028+0.076+0.149+0.212=0.473

Therefore, the probability that at least 6 wolves were females is 0.473.

Probability that less than 4 wolves were females:

Less than 4 females are the same as more than 8 males.

The probability that less than 4 wolves were females is calculated as given below:

P(r>8)=P(9)+P(10)+P(11)+P(12)=C12,9(0.55)9(0.45)3+C12,10(0.55)10(0.45)2+C12,11(0.55)11(0.45)1+C12,12(0.55)12(0.45)0=0.092+0.034+0.008+0.001=0.135

Therefore, the probability that less than 4 wolves were females is 0.135.

b.

To determine

Compute the probability that at least 6 wolves were males, at least 6 wolves were females, and less than 4 wolves were females from 1918 to present.

b.

Expert Solution
Check Mark

Answer to Problem 17P

The probability that at least 6 wolves were males is 0.961.

The probability that at least 6 wolves were females is 0.117.

The probability that less than 4 wolves were females is 0.493.

Explanation of Solution

Calculation:

A trial determines the sex of a wolf with two possible outcomes success or failure. Consider success as male and failure as female.

The total number of wolves (or trials) is n=12.

The probability of male wolves (or success) is p=0.70.

The probability of female wolves (or failure) is q=0.30.

Probability that at least 6 wolves were males:

The probability that at least 6 wolves were males is calculated as given below:

P(r6)=P(6)+P(7)+P(8)+P(9)+P(10)+P(11)+P(12)=C12,6(0.70)6(0.30)6+C12,7(0.70)7(0.30)5+C12,8(0.70)8(0.30)4+C12,9(0.70)9(0.30)3+C12,10(0.70)10(0.30)2+C12,11(0.70)11(0.30)1+C12,12(0.70)12(0.30)0=0.079+0.158+0.231+0.240+0.168+0.071+0.014=0.961

Therefore, the probability that at least 6 wolves were males is 0.961.

Probability that at least 6 wolves were females:

Six or more females are the same as less than or equal to males.

The probability that at least 6 wolves were females is calculated as given below:

P(r6)=P(0)+P(1)+P(2)+P(3)+P(4)+P(5)+P(6)=C12,0(0.70)0(0.30)12+C12,1(0.70)1(0.30)11+C12,20.702(0.30)10+C12,3(0.70)3(0.30)9+C12,4(0.70)4(0.30)8+C12,5(0.70)5(0.30)7+C12,6(0.70)6(0.30)6=0.000+0.000+0.000+0.001+0.008+0.029+0.079=0.117

Therefore, the probability that at least 6 wolves were females is 0.117.

Probability that less than 4 wolves were females:

Less than 4 females are the same as more than 8 males.

The probability that less than 4 wolves were females is calculated as given below:

P(r>8)=P(9)+P(10)+P(11)+P(12)=C12,9(0.70)9(0.30)3+C12,10(0.70)10(0.30)2+C12,11(0.70)11(0.30)1+C12,12(0.70)12(0.30)0=0.240+0.168+0.071+0.014=0.493

Therefore, the probability that less than 4 wolves were females is 0.493.

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Chapter 5 Solutions

Understandable Statistics: Concepts and Methods

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