Question

Use the following information to answer the next four exercises. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about$4\%$. Of the next $25$ patients calling in claiming to have the flu, we are interested in how many actually have the flu.

Find the probability that at least four of the $25$patients actually have the flu.

Short answer

The chance of at least four out of every $25$ patients getting the flu is $0.0165.$

Step-by-step solution

Given information

The total number of people polled was $25.$

The probability of being chosen and having flu is $0.04.$

Explanation

$$\begin{gathered} =P\left(X\ge 4\right) \\ =1-P\left(X<4\right) \\ =1-\left[P\left(X=0\right)+P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)\right] \\ =1-\left[\frac{25!}{0!\left(25-0\right)!}\times {\left(0.04\right)}^0+\frac{25!}{1!\left(25-1\right)!}\times {\left(0.04\right)}^1+\frac{25!}{2!\left(25-2\right)!}\times {\left(0.04\right)}^2+\frac{25!}{3!\left(25-3\right)!}\times {\left(0.04\right)}^3\right] \\ =0.0165 \end{gathered}$$

Because finding that manually more than equal to will take a long time and effort, we deduct the likelihood that X is less than $3$ because the sum of all probabilities for a random variable is$1.$

The chance of at least four out of every $25$patients getting the flu is $0.0165.$