Question

What is the probability that a waitress will
refuse to serve alcoholic beverages to only 2 minors
if she randomly checks the IDs of 5 among 9 students,
4 of whom are minors?

Answer 1

Answer on Question #72622, Math / Statistics and Probability

What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the IDs of 5 among 9 students, 4 of whom are minors?

Solution

Let $X$ be the random variable which denotes the number of minors among the 5 students selected at random for ID checking.

The total number of students $N = 9$.

The number of students who are not of legal age (minor) $k = 4$.

Hence, $X$ has a hypergeometric distribution with parameters $N = 9, n = 5$ and $k = 4$.

$$X \sim HyperGeom(N, k, n)$$

$p.m.f$ of $X$ is given by

$$h(x; N = 9, n = 5, k = 4) = \frac{\binom{k}{x} \binom{N - k}{n - x}}{\binom{N}{n}},$$

where $\max\{0, n - (N - k)\} \leq x \leq \min\{n, k\}$

$$i.e. 0 \leq x \leq 4$$

The probability that a waitress will refuse to serve alcoholic beverages to only 2 minors

$$\begin{array}{l} P(X = 2) = \frac{\binom{4}{2} \binom{9 - 4}{5 - 2}}{\binom{9}{5}} = \frac{\binom{4}{2} \binom{5}{3}}{\binom{9}{5}} = \frac{\frac{4!}{2! (4 - 2)!} \cdot \frac{5!}{3! (5 - 3)!}}{\frac{9!}{5! (9 - 5)!}} = \\ = \frac{\frac{4(3)}{1(2)} \cdot \frac{4(5)}{1(2)}}{\frac{9(8)(7)(6)}{1(2)(3)(4)}} = \frac{6(10)}{18(7)} = \frac{10}{21} \approx 0.47619 \end{array}$$

Answer: $\frac{10}{21} \approx 0.47619$.

Answer provided by AssignmentExpert.com

Step-by-Step Solution:

Step 1 of 3

No. of students who are not of legal age(minor) = 4

total no. of students = 9

Let $X$ be the random variable which denotes the no. of minorss among the 5

students selected at random for ID checking

$\therefore X$ has a hypergeometric distribution with parameters $N = 9$, $n = 5$ and $k = 4$

and p.m.f of $X$ is given by

$$h(x; N = 9, n = 5, k = 4) = \frac{\binom{k}{x} \binom{N - k}{n - x}}{\binom{N}{n}}$$

where $\max \{0, n - (N - k)\} \leq x \leq \min \{n, k\}$

i.e, $0 \leq x \leq 4$

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