Chapter 6.2, Problem 48PS

Solution

a)

To construct: The histogram for the histogram that depicts the number who came to visit an art museum

Given information:

  $\begin{array}{c}\text{Probability of success}\left(p\right)=0.35\\ \text{Number of trials}\left(n\right)=10\end{array}$

Formula used:

The pdf of the binomial distribution is:

  $P\left(X=x\right){=}^n{C}_x\times p^x\times {\left(1-p\right)}^{n-x}$

Where,

n is the number of trials.

p is the probability of success.

x is the number of successes.

Graph:

The histogram is constructed as:

  

b)

To find: The probability that at most 4 people has visited to the museum.

The probability is 0.75150.

Calculation:

Consider, X be the random variable that follows the binomial distribution.

The probability is calculated as:

  $\begin{array}{c}P\left(X\le 4\right)=P\left(X=0\right)+P\left(X=1\right)+P\left(X=2\right)+P\left(X=3\right)+P\left(X=4\right)\\ =\left[\left({}^{10}{C}_0\times {\left(0.35\right)}^{10}\times {\left(1-0.35\right)}^{10-0}\right)+\mathrm{.........}+\left({}^{10}{C}_4\times {\left(0.35\right)}^4\times {\left(1-0.35\right)}^{10-4}\right)\right]\\ =0.01346+0.07249+0.17565+0.25222+0.25222+0.23767\\ =0.75150\end{array}$

The required probability is 0.75150.