Chapter 15, Problem 24E

To determine

To Explain: that the species and sex of the animals are independent.

Answer to Problem 24E

Thus species and sex of the animal are independent.

Explanation of Solution

Given:

There are 24 dogs and 18 cats available for adoption.

Total number of males dogs = 8

Total number of Males cats = 6

Formula used:

  $P\left(D|M\right)=\frac{P\left(D\cap M\right)}{P\left(M\right)}$

Calculation:

Following table mention description of animal by gender and species.

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

D = animal is dog

C = animal is cat

M = animal is male

  $P\left(D|M\right)=\frac{P\left(D\cap M\right)}{P\left(M\right)}$

  $\begin{array}{c}P\left(D\cap M\right)=\frac{8}{42}\\ P\left(M\right)=\frac{14}{42}\end{array}$

Putting the values

  $\begin{array}{c}P\left(D\backslash M\right)=\frac{\frac{8}{42}}{\frac{14}{42}}\\ =\frac{8}{42}\times \frac{42}{14}\\ =\frac{8}{14}\\ P\left(D\right)=\frac{24}{42}\\ =\frac{8}{14}\\ P\left(D\backslash M\right)=P\left(D\right)\end{array}$

Thus species and sex of the animal are independent.