Chapter 6, Problem 11T

Solution

To Calculate:

The probability of $k$ successes for a binomial experiment consisting of $n$ trials with probability $p$ of success on each trial .

  $0.7748$

Given:

  $k\ge 8,n=9,p=0.9$

Concepts Used:

In a binomial experiment with $n$ independent trials each having probability of success $p$ . The probability of $r$ successes is given by ${}_n{C}_r{p}^r{\left(1-p\right)}^{n-r}$ .

The probability of occurrence of the union of many events that are mutually exclusive is the sum of their individual probabilities.

The definition ${}_n{C}_r=\frac{n!}{r!\left(n-r\right)!}$ .

The probability of occurrence of an event and that of its complement add up to $1$ .

Calculations:

The probability of success on each trial is $p=0.9$ . Calculate the probability of at least $8$ successes among $9$ trials ( $k\ge 8,n=9$ ).

  $\begin{array}{c}P\left(\text{at least 8 successes}\right)=P\left(\text{8 successes}\right)+P\left(\text{9 successes}\right)\\ {=}_9{C}_8{\left(0.9\right)}^8{\left(0.1\right)}^1{+}_9{C}_9{\left(0.9\right)}^9{\left(0.1\right)}^0\\ ={\left(0.9\right)}^8\left(9\times 0.1+0.9\right)\\ \approx 0.7748\end{array}$

Conclusion:

  $0.7748$ is the required probability.