Question

it is known that a certain prescription drug produces undesirable side effects in 37% of all patients who use it. among a random sample of eight patients using the drug, find the probability that more than two have undesirable side effects.
the probability that, among the eight patients, more than two have undesirable side effects is
(round to three decimal places as needed.)

Explanation:

Step1: Identify the binomial distribution parameters

Let \(n = 8\) (number of trials/sample size), \(p=0.37\) (probability of success - having side - effects), and \(q = 1 - p=1 - 0.37 = 0.63\). The binomial probability formula is \(P(X = k)=C(n,k)\times p^{k}\times q^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\).

Step2: Calculate \(P(X\leqslant2)\)

\[

$$\begin{align*} P(X = 0)&=C(8,0)\times(0.37)^{0}\times(0.63)^{8}\\ &=1\times1\times(0.63)^{8}\\ &\approx0.0168 \end{align*}$$

\]
\[

$$\begin{align*} P(X = 1)&=C(8,1)\times(0.37)^{1}\times(0.63)^{7}\\ &=8\times0.37\times(0.63)^{7}\\ &\approx0.0907 \end{align*}$$

\]
\[

$$\begin{align*} P(X = 2)&=C(8,2)\times(0.37)^{2}\times(0.63)^{6}\\ &=\frac{8!}{2!(8 - 2)!}\times(0.37)^{2}\times(0.63)^{6}\\ &=28\times0.1369\times(0.63)^{6}\\ &\approx0.2142 \end{align*}$$

\]
\[P(X\leqslant2)=P(X = 0)+P(X = 1)+P(X = 2)\approx0.0168 + 0.0907+0.2142=0.3217\]

Step3: Calculate \(P(X>2)\)

\[P(X>2)=1 - P(X\leqslant2)=1 - 0.3217 = 0.6783\approx0.678\]

Answer:

\(0.678\)