Question

zacks city is always full of tourists, so many cars display out - of - state license plates. on his walk to work, he passed 12 parked cars, 7 of which had arizona plates. if zack checks the plates on 9 randomly selected cars, what is the probability that exactly 6 of the chosen cars had arizona plates? write your answer as a decimal rounded to four decimal places.

Explanation:

Step1: Identify combination formula

We use the hyper - geometric probability formula $P(X = k)=\frac{\binom{M}{k}\binom{N - M}{n - k}}{\binom{N}{n}}$, where $N$ is the total number of items, $M$ is the number of "success" items in the total, $n$ is the number of items selected, and $k$ is the number of "success" items in the selected group. Here, $N = 12$, $M=7$, $n = 9$, $k = 6$.

Step2: Calculate binomial coefficients

$\binom{M}{k}=\binom{7}{6}=\frac{7!}{6!(7 - 6)!}=\frac{7!}{6!1!}=7$
$\binom{N - M}{n - k}=\binom{12-7}{9 - 6}=\binom{5}{3}=\frac{5!}{3!(5 - 3)!}=\frac{5\times4\times3!}{3!\times2\times1}=10$
$\binom{N}{n}=\binom{12}{9}=\binom{12}{3}=\frac{12!}{3!(12 - 3)!}=\frac{12\times11\times10\times9!}{3!\times9!}=220$

Step3: Calculate the probability

$P(X = 6)=\frac{\binom{7}{6}\binom{5}{3}}{\binom{12}{9}}=\frac{7\times10}{220}=\frac{70}{220}\approx0.3182$

Answer:

$0.3182$