Chapter 11, Problem 19SGR

To determine

To find: The given experiment is a binomial experiment or not and hence find the probability.

Answer to Problem 19SGR

Theexperiment can be reduced to a binomial experiment, and the probability is $34.3\%$ .

Explanation of Solution

The correct answer is given as:

As in a binomial experiment each trial has only two possible outcomes, success or failure.

Since in the given experiments there are two possibilities are arising as either a person’s favourite season is summer or not.

Therefore, the given experiment can be reduced to a binomial experiment.

The probability of $X$ successes in $n$ independent trials is given as:

  $P\left(X\right)=C{}_X{p}^X{q}^{n-X}$

Here,

  $\begin{array}{l}n=20\\ X=15\\ p=0.68\\ q=0.32\end{array}$

Further simplified for the probability as:

  $\begin{array}{c}P\left(X\ge 15\right)=P\left(15\right)+P\left(16\right)+P\left(17\right)+P\left(18\right)+P\left(19\right)+P\left(20\right)\\ =C{}_{15}{\left(0.68\right)}^{15}{\left(0.32\right)}^5+C{}_{16}{\left(0.68\right)}^{16}{\left(0.32\right)}^4+C{}_{17}{\left(0.68\right)}^{17}{\left(0.32\right)}^3\\ +C{}_{18}{\left(0.68\right)}^{18}{\left(0.32\right)}^2+C{}_{19}{\left(0.68\right)}^{19}{\left(0.32\right)}^1+C{}_{20}{\left(0.68\right)}^{20}{\left(0.32\right)}^0\\ \approx 0.343\\ =34.3\%\end{array}$