Question

a certain prescription drug is known to produce undesirable side effects in 30% of all patients who use the drug. among a random sample of six patients, find the probability of the stated event. exactly three have undesirable side effects. the probability of three patients having undesirable side effects, among a random sample of six is (simplify your answer. type an integer or a decimal rounded to three decimal places as needed.)

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 $C(n,k)=\frac{n!}{k!(n - k)!}$.

Step2: Determine values of n, k, and p

Here, $n = 6$ (number of patients in the sample), $k = 3$ (number of patients with side - effects), and $p=0.3$ (probability of a patient having side - effects), $1 - p = 0.7$.

Step3: Calculate the combination C(n,k)

$C(6,3)=\frac{6!}{3!(6 - 3)!}=\frac{6!}{3!3!}=\frac{6\times5\times4}{3\times2\times1}=20$.

Step4: Calculate the probability

$P(X = 3)=C(6,3)\times(0.3)^{3}\times(0.7)^{6 - 3}=20\times0.027\times0.343 = 0.18522\approx0.185$.

Answer:

$0.185$