Chapter 6.2, Problem 28E

Solution

To find: the probability of $k$ successes.

  $P\left(k\le 3\right)\approx 0.874$

Given:

  $k\le 3,n=7,p=0.3$.

Concept used:

For a binomial experiment, the probability of $k$ successes in $n$ trails is,

  $P\left(k\text{successes}\right)=C{}_k{p}^k{\left(1-p\right)}^{n-k}$ , where $p$ is the probability of success.

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

Calculation:

  $\begin{array}{c}P\left(k\le 3\right)=P\left(k=3\right)+P\left(k=2\right)+P\left(k=1\right)+P\left(k=0\right)\\ =C{}_3{\left(0.3\right)}^3{\left(1-0.3\right)}^{7-3}+C{}_2{\left(0.3\right)}^2{\left(1-0.3\right)}^{7-2}+C{}_1{\left(0.3\right)}^1{\left(1-0.3\right)}^{7-1}+C{}_0{\left(0.3\right)}^0{\left(1-0.3\right)}^{7-0}\\ =35{\left(0.3\right)}^3{\left(0.7\right)}^4+21{\left(0.3\right)}^2{\left(0.7\right)}^5+7\left(0.3\right){\left(0.7\right)}^6+{\left(0.7\right)}^7\\ \approx 0.874\end{array}$