According to the National Traffic Safety Council of Namibia, the probability that a traffic fatality will involve an intoxicated or alcohol-impaired driver is 40%. If eight traffic fatalities observed last month.

a. Find the probability that the number of an intoxicated or alcohol-impaired driver is exactly three.
b. Find the expected value and standard deviation of the number of intoxicated or alcohol-impaired drivers.

Solution

Let $X$ denote the number of an intoxicated or alcohol-impaired driver.

Let $p=0.4$ be the probability that a traffic fatality will involve an intoxicated or alcohol-impaired driver. Then the random variable $X$ follows Binomial distribution, i.e., $X\sim B(8,0.4)$.

The probability mass function of $X$ is

$$ \begin{aligned} P(X=x) &= \binom{8}{x} (0.4)^x (1-0.4)^{8-x}, \\ &\quad x=0,1,\cdots, 8. \end{aligned} $$

a. The probability that the number of an intoxicated or alcohol-impaired driver is exactly three

$$ \begin{aligned} P(X= 3) & =\binom{8}{3} (0.4)^{3} (1-0.4)^{8-3}\\ & = 0.2787\\ \end{aligned} $$

b. The expected value and standard deviation of the number of intoxicated or alcohol-impaired drivers is

$$ \begin{aligned} E(X) &= np\\ &= 8 \times 0.4\\ &= 3.2 \end{aligned} $$

The standard deviation of the number of intoxicated or alcohol-impaired drivers is

$$ \begin{aligned} sd(X) &= \sqrt{np(1-p)}\\ &= \sqrt{8 \times 0.4\times (1-0.4)}\\ &= 1.3856 \end{aligned} $$

Further Reading