Chapter 11, Problem 3PT

(a)

To determine

To find: The probability that there will be exactly 12 successes.

Answer to Problem 3PT

The probability that there will be exactly 12 successes is $11.1\%$ .

Explanation of Solution

Given:

A binomial distribution has a $65\%$ rate of success.

The numbers of trials are 15.

Calculation:

The probability of $X$ successes in $n$ independent trials is given by $P\left(X\right){=}_n{C}_X{p}^X{q}^{n-X}$ .

Find the probability that there will be exactly 12 successes as follows.

  $\begin{array}{c}P\left(12\right){=}_{15}{C}_{12}{\left(0.65\right)}^{12}{\left(0.35\right)}^3\\ \approx 0.111\\ \approx 11.1\%\end{array}$

Therefore, the probability that there will be exactly 12 successes is $11.1\%$ .

(b)

To determine

To find: The probability that there will be at least 10 successes.

Answer to Problem 3PT

The probability that there will be at least 10 successes is $56.4\%$ .

Explanation of Solution

Given:

A binomial distribution has a $65\%$ rate of success.

The numbers of trials are 15.

Calculation:

The probability of $X$ successes in $n$ independent trials is given by $P\left(X\right){=}_n{C}_X{p}^X{q}^{n-X}$ .

Find the probability that there will be at least 10 successes as follows.

  $\begin{array}{c}P\left(X\ge 10\right)=P\left(10\right)+P\left(11\right)+P\left(12\right)+P\left(13\right)+P\left(14\right)+P\left(15\right)\\ {=}_{15}{C}_{10}{\left(0.65\right)}^{10}{\left(0.35\right)}^5{+}_{15}{C}_{11}{\left(0.65\right)}^{11}{\left(0.35\right)}^4{+}_{15}{C}_{12}{\left(0.65\right)}^{12}{\left(0.35\right)}^3{+}_{15}{C}_{13}{\left(0.65\right)}^{13}{\left(0.35\right)}^2{+}_{15}{C}_{14}{\left(0.65\right)}^{14}{\left(0.35\right)}^1{+}_{15}{C}_{15}{\left(0.65\right)}^{15}{\left(0.35\right)}^0\\ \approx 0.564\\ \approx 56.4\%\end{array}$

Therefore, the probability that there will be at least 10 successes is $56.4\%$ .