Question

find the indicated probability. a report from the secretary of health and human services stated that 70% of single - vehicle traffic fatalities that occur at night on weekends involve an intoxicated driver. if a random sample of 15 single - vehicle traffic fatalities that occur at night on a weekend is selected, find the probability that 11 involve a driver who is intoxicated. 0.57 0.22 0.39 0.69 0.71

Explanation:

Step1: Identify binomial formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$.
Here, $n = 15$, $k = 11$, and $p=0.7$.

Step2: Calculate the combination $C(n,k)$

$C(15,11)=C(15,4)=\frac{15!}{4!(15 - 4)!}=\frac{15!}{4!×11!}=\frac{15\times14\times13\times12}{4\times3\times2\times1}=1365$.

Step3: Calculate $p^{k}$ and $(1 - p)^{n - k}$

$p^{k}=(0.7)^{11}$ and $(1 - p)^{n - k}=(0.3)^{4}$.
$(0.7)^{11}\approx0.0197732674$ and $(0.3)^{4}=0.0081$.

Step4: Calculate the probability

$P(X = 11)=C(15,11)\times(0.7)^{11}\times(0.3)^{4}$
$P(X = 11)=1365\times0.0197732674\times0.0081\approx0.22$.

Answer:

0.22