Chapter 6, Problem T6.8SPT

To determine

To find: The correct answer choice.

Answer to Problem T6.8SPT

The correct answer choice is (b), 0.035.

Explanation of Solution

Given:

  $\begin{array}{l}\text{Number of people that are exposed to disease}\left(n\right)=17\\ \text{Probabilty of contracting the disease}\left(p\right)=40\%=0.40\end{array}$

The provided options are:

1. 0.011
2. 0.035
3. 0.092
4. 0.965
5. 0.989

Formula used:

The formula to compute the probability in binomial distribution is:

  $P\left(X=x\right)=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{\left(1-p\right)}^{n-x}$

Calculation:

The probability that hospital capacity will be exceeded can be calculated as:

  $\begin{array}{c}P\left(X>10\right)={\displaystyle \sum _{11}^{17}\left(\begin{array}{l}17\\ x\end{array}\right){\left(0.40\right)}^x{\left(1-0.40\right)}^{17-x}}\\ =\left[P\left(X=11\right)+P\left(X=12\right)+\mathrm{....}+P\left(X=17\right)\right]\\ =\left[\begin{array}{l}\left(\left(\begin{array}{l}17\\ 11\end{array}\right){\left(0.40\right)}^{11}{\left(1-0.40\right)}^{17-11}\right)+\left(\left(\begin{array}{l}17\\ 12\end{array}\right){\left(0.40\right)}^{12}{\left(1-0.40\right)}^{17-12}\right)+\mathrm{...}+\\ \left(\left(\begin{array}{l}17\\ 12\end{array}\right){\left(0.40\right)}^{17}{\left(1-0.40\right)}^{17-17}\right)\end{array}\right]\\ =0.035\end{array}$

Hence, the correct option is (b), 0.035.