Question

according to a recent survey, 81 percent of adults in a certain state have graduated from high school. if 15 adults from the state are selected at random, what is the probability that 5 of them have not graduated from high school?

Explanation:

Step1: Define success probability

Probability of not graduating: $p = 1 - 0.81 = 0.19$

Step2: Identify binomial parameters

$n=15$, $k=5$, $p=0.19$, $q=1-p=0.81$

Step3: Apply binomial formula

Binomial probability: $P(X=k)=\binom{n}{k}p^kq^{n-k}$
$$\binom{15}{5}(0.19)^5(0.81)^{10}$$

Step4: Calculate combination term

$\binom{15}{5}=\frac{15!}{5!(15-5)!}=3003$

Step5: Compute individual terms

$(0.19)^5\approx0.0002476$, $(0.81)^{10}\approx0.1215767$

Step6: Multiply all terms

$3003\times0.0002476\times0.1215767\approx0.090$

Answer:

$\approx0.090$ (or 9.0% when expressed as a percentage)