Question

In a litter of seven kittens, three are female.
You pick two kittens at random.
a. Create a probability model for the
number of male kittens you get.
b. Find the expected number of males.
c. Find the standard deviation for your
distribution.

Answer 1

Answer on Question #46950 – Math – Statistics and Probability

Question:

In a litter of seven kittens, three are female. You pick two kittens at random.

a. Create a probability model for the number of male kittens you get.

b. Find the expected number of males.

c. Find the standard deviation for your distribution.

Solution:

a. There are total $\binom{7}{2} = 21$ ways to pick 2 kitties from 7.

We can pick 0, 1, or 2 male kittens. Denote the random variable of male kitties picked as X.

$$P(X = 0) = P(\text{pick 2 female kittens}) = \frac{\binom{3}{2}}{21} = \frac{3}{21} = \frac{1}{7}.$$ $$P(X = 1) = P(\text{pick 1 male and 1 female kitten}) = \frac{\binom{3}{1}\binom{4}{1}}{21} = \frac{3 \cdot 4}{21} = \frac{4}{7}.$$ $$P(X = 2) = P(\text{pick 2 male kittens}) = \frac{\binom{4}{2}}{21} = \frac{6}{21} = \frac{2}{7}.$$

So, for X:

b. The expected number of males is $E(X) = 0 \cdot \frac{1}{7} + 1 \cdot \frac{4}{7} + 2 \cdot \frac{2}{7} = \frac{8}{7}$.

c. $E(X^2) = 0 \cdot \frac{1}{7} + 1 \cdot \frac{4}{7} + 4 \cdot \frac{2}{7} = \frac{12}{7}$. Then

The standard deviation is $\sigma(X) = (E(X^2) - E(X)^2)^{1/2} = \left(\frac{12}{7} - \frac{64}{49}\right)^{1/2} = \left(\frac{20}{49}\right)^{1/2} = \frac{2\sqrt{5}}{7}$.

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