Chapter 11.4, Problem 40PPS

To determine

To find the probability of $s$ successes.

Answer to Problem 40PPS

  $\boxed{0.474}$ Or $\boxed{47.4\%}$

Explanation of Solution

Given information:

Each binomial distribution has $n$ trials and $p$ probabilities of success. Determine the probability of $s$ successes.

  $n=10,p=0.75,s\ge 8$

Calculation:

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

  $p\left(k\right)=C{}_k{\left(p\right)}^k{\left(q\right)}^{n-k}$

Thus, $p$ represents the probability of success and $q$ represents the probability of failure.

Now, consider a binomial probability with the following attributes:

  $p=0.75,n=10$

In the above expression, the probability of failure is,

  $\begin{array}{l}q=1-p\\ =1-0.75\\ =0.25\end{array}$

Now, we will consider the probability that there are more than or equal to $8$ successes out of $10$ trials. So, there could be $8,9$ or $10$ successes.

Thus, the probability that there are more than or equal to $8$ successes out of $10$ trials is,

  $p\left(k\ge 8\right)=p\left(8\right)+p\left(9\right)+p\left(10\right)$

Now, substitute the above $k$ values in the above expression,

  $p\left(k\ge 8\right)=C{}_8{\left(0.75\right)}^8{\left(0.25\right)}^{10-8}+C{}_9{\left(0.75\right)}^9{\left(0.25\right)}^{10-9}+C{}_{10}{\left(0.75\right)}^{10}{\left(0.25\right)}^{10-10}$

  $=1-\left(0.282+0.188+0.056\right)$

  $=0.474$

Hence, the value of $p\left(k\ge 8\right)$ for $p=0.75,n=10$ is $\boxed{0.474}$ or $\boxed{47.4\%}$ .