Question

Male nannies. According to the International Nanny Association (INA), 4,176 nannies were placed in a job during a recent year (www.nanny.org). Of these, only 24 were men. In Exercise 3.12 (p. 169) you found the probability that a randomly selected nanny who was placed during a recent year is a man. Now use the hypergeometric distribution to find the probability that in a random sample of 10 nannies who were placed during a recent year, at least 1 is a man

Short answer

The probability that among 10 nannies were placed, at least 1 is a man is 0.057.

Step-by-step solution

Given Information

The total number of nannies were placed in a job is

N = 4176

Number of men nannies is

M = 24

The sample size is n = 10

To find the probability of male nanny

$$\begin{gathered} p\left(\text{male} \text{nanny}\right)=\frac{M}{N} \\ =\frac{24}{4176} \\ =0.0057 \end{gathered}$$

Therefore, the probability of male nanny is 0.0057.

Compute the probability

A minimum of two steps are required.

Here, we use hypergeometric distribution to finding the probability,

$$\begin{gathered} p\left(x=0\right)=\frac{\left(24 \\ 0\right)\left(4176-24 \\ 10-0\right)}{\left(4176 \\ 10\right)} \\ =\frac{\left(24 \\ 0\right)\left(4152 \\ 10\right)}{\left(4176 \\ 10\right)} \\ =0.943 \end{gathered}$$

To get the required probability,

$$\begin{gathered} p\left(x⩾1\right)=1-p\left(x=0\right) \\ =1-0.943 \\ =0.057 \end{gathered}$$

Therefore, the probability is 0.057.