Chapter 11.4, Problem 17PPS

To determine

Find the probability that they win $7$ of their next $12$ games.

Answer to Problem 17PPS

  $\boxed{0.0956or9.56\%}$

Explanation of Solution

Given information:

A certain football team has won $75.7\%$ of their games. Find the probability that they win $7$ of their next $12$ games.

Calculation:

Consider a football team that wins $75.7\%$ of the matches they have played so far.

The probability of $k$ successes in $n$ trials of a binomial experiment is,

  $P(k){=}^n{C}_k{(p)}^k{(q)}^{n-k}$

Here $p$ represents the probability of success and $q$ represents the probability of failure which is equal to,

  $q=1-p$

Since the football team that wins $75.7\%$ of the matches they have played so far, so,

  $\begin{array}{l}p=75.7\%\\ =0.757\end{array}$

The probability of failure is,

  $q=1-p$

  $=1-0.757$

  $\begin{array}{l}=0.243\\ \end{array}$

Substitute $p=0.757,q=0.243,k=7,n=12$ in the above formula to find the probability that the football team will win $7$ out of the next $12$ matches,

  $\begin{array}{l}P(7){=}^{12}{C}_7{(}^{0.757}{(}^{0.243}\\ =792{(}^{0.757}{(}^{0.243}\\ =0.0956\\ =9.56\%\end{array}$

Hence, the probability that the football team will win $7$ out of the next $12$ matches is $\boxed{0.0956or9.56\%}$