Chapter 15, Problem 8E

(a)

To determine

To find: the conditional probability if an animal is chosen at arbitrary.

Answer to Problem 8E

33.33%

Explanation of Solution

Given:

The pet is male and given that it is a cat.

	Male	Female	Total
Dogs	8	24-8=16	24
Cats	6	18-6=12	18
Total	14	28	42

Formula used:

  $P\left(M/C\right)=\frac{P\left(M\cap C\right)}{P\left(C\right)}$

Calculation:

Suppose M is the event that pet is male and C is the event that is cat.

Probability the table there are 6 male cats

  $P\left(M/C\right)=\frac{P\left(M\cap C\right)}{P\left(C\right)}$

From the table, there are 6 male cats

  $\begin{array}{c}P\left(M\cap C\right)=\frac{6}{42}\\ P\left(C\right)=\frac{18}{42}\end{array}$

Now, plug in the value

  $\begin{array}{c}P\left(M/C\right)=\frac{\frac{6}{42}}{\frac{18}{42}}\\ =0.3333\end{array}$

There is 33.33% possibility that pet is male given that cat is chosen.

(b)

To determine

To find: the conditional probability if an animal is chosen at arbitrary.

Answer to Problem 8E

42.86%

Explanation of Solution

Given:

The pet is a cat and given that it is female.

	Male	Female	Total
Dogs	8	24-8=16	24
Cats	6	18-6=12	18
Total	14	28	42

Formula used:

  $P\left(C/F\right)=\frac{P\left(C\cap F\right)}{P\left(F\right)}$

Calculation:

Suppose F is the event that pet is female and C is the event that pet is cat.

Probability that pet is cat given that it is female is,

  $P\left(C/F\right)=\frac{P\left(C\cap F\right)}{P\left(F\right)}$

From the table,

  $\begin{array}{c}P\left(C\cap F\right)=\frac{12}{42}\\ P\left(F\right)=\frac{28}{42}\end{array}$

Now, putting the value

  $\begin{array}{c}P\left(C/F\right)=\frac{\frac{12}{42}}{\frac{28}{42}}\\ =0.4286\end{array}$

There is 42.86% possibility that pet is cat given that is female.

(c)

To determine

To find: the conditional probability if an animal is chosen at arbitrary.

Answer to Problem 8E

66.67%

Explanation of Solution

Given:

The pet is female and given that it is a dog.

	Male	Female	Total
Dogs	8	24-8=16	24
Cats	6	18-6=12	18
Total	14	28	42

Formula used:

  $P\left(F/D\right)=\frac{P\left(F\cap D\right)}{P\left(D\right)}$

Calculation:

Suppose F is the event that pet is female and D is the event that pet is dog.

Probability that pet is female given that it is dog is,

  $P\left(F/D\right)=\frac{P\left(F\cap D\right)}{P\left(D\right)}$

From the table,

  $\begin{array}{c}P\left(F\cap D\right)=\frac{16}{42}\\ P\left(D\right)=\frac{24}{42}\end{array}$

Now, putting the values

  $\begin{array}{c}P\left(F/D\right)=\frac{\frac{16}{42}}{\frac{24}{42}}\\ =\frac{16}{24}\\ =0.6667\end{array}$

There is 66.67% possibility that pet is female given that pet is dog.