Chapter 7, Problem 1P

a.

To determine

Compute the probability that “All 15 will show significant improvement”.

Answer to Problem 1P

The probability that “All 15 will show significant improvement” is 0.035.

Explanation of Solution

The formula to find the binomial probability is as follows:

$\begin{array}{l}P\left(X\right)=C{}_x{\pi }^x{\left(1-\pi \right)}^{\left(n-x\right)}\\ \text{where, }n\text{ is the number of trials}\text{.}\\ x\text{ is the random variable}\text{.}\\ \pi \text{ is the probability of success}\text{.}\end{array}$

The distribution of significant improvement of new medicine for acne follows Binomial distribution with sample size of people n as 15 and probability that of effectiveness, π is 0.80.

Let the random variable x be the number of people who show significant improvement.

The given probability is calculated as follows:

Here, n = 15, x = 15 and π = 0.80.

$\begin{array}{c}P\left(x=15\right)=C{}_{15}{\left(0.80\right)}^{15}{\left(1-0.80\right)}^{\left(15-15\right)}\\ =\frac{15!}{15!\left(15-15\right)!}\times {0.80}^{15}\times {0.20}^0\\ =1\times 0.035\times 1\\ =0.035\end{array}$

Therefore, the probability that “All 15 will show significant improvement” is 0.035.

b.

To determine

Compute the probability that ““Fewer than 9 of 15 will show significant improvement”.

Answer to Problem 1P

The probability that “Fewer than 9 of 15 will show significant improvement” is 0.018.

Explanation of Solution

The probability that “Fewer than 9 of 15 will show significant improvement” is calculated as follows:

Here, n = 15, x = 9 and π = 0.80.

$\begin{array}{c}P\left(x<9\right)=\left[\begin{array}{l}P\left(x=0\right)+P\left(x=1\right)+P\left(x=2\right)\\ +\mathrm{....}+P\left(x=8\right)\end{array}\right]\\ =\left\{\begin{array}{l}\left[C{}_0{\left(0.80\right)}^0{\left(1-0.80\right)}^{\left(15-0\right)}\right]+\left[C{}_1{\left(0.80\right)}^1{\left(1-0.80\right)}^{\left(15-1\right)}\right]\\ +\left[C{}_2{\left(0.80\right)}^2{\left(1-0.80\right)}^{\left(15-2\right)}\right]\\ +\mathrm{....}+\left[C{}_8{\left(0.80\right)}^8{\left(1-0.80\right)}^{\left(15-8\right)}\right]\end{array}\right\}\\ =\left[\begin{array}{l}\left(\frac{15!}{0!\left(15-0\right)!}\times {0.80}^0\times {0.20}^{15}\right)+\left(\frac{15!}{1!\left(15-1\right)!}\times {0.80}^1\times {0.20}^{14}\right)\\ +\left(\frac{15!}{2!\left(15-2\right)!}\times {0.80}^2\times {0.20}^{13}\right)\\ +\mathrm{.....}+\left(\frac{15!}{8!\left(15-8\right)!}\times {0.80}^8\times {0.20}^7\right)\end{array}\right]\\ =\left(0.0000+0.0000+0.0000+\mathrm{.....}+0.0138\right)\end{array}$              

               $=0.018$

Therefore, the probability that “Fewer than 9 of 15 will show significant improvement” is 0.018.

c.

To determine

Compute the probability that “12 or more people will show significant improvement”.

Answer to Problem 1P

The probability that “12 or more people will show significant improvement” is 0.648.

Explanation of Solution

The probability that 12 or more people will show significant improvement is calculated as follows:

Here, n=15, x=12 and π=0.80.

$\begin{array}{c}P\left(x\ge 12\right)=\left[\begin{array}{l}P\left(x=12\right)+P\left(x=13\right)+P\left(x=14\right)\\ +P\left(x=15\right)\end{array}\right]\\ =\left\{\begin{array}{l}\left[C{}_{12}{\left(0.80\right)}^{12}{\left(1-0.80\right)}^{\left(15-12\right)}\right]+\left[C{}_{13}{\left(0.80\right)}^{13}{\left(1-0.80\right)}^{\left(15-13\right)}\right]\\ +\left[C{}_{14}{\left(0.80\right)}^{14}{\left(1-0.80\right)}^{\left(15-14\right)}\right]+\left[C{}_{15}{\left(0.80\right)}^{15}{\left(1-0.80\right)}^{\left(15-15\right)}\right]\end{array}\right\}\\ =\left[\begin{array}{l}\left(\frac{15!}{12!\left(15-12\right)!}\times {0.80}^{12}\times {0.20}^3\right)+\left(\frac{15!}{13!\left(15-13\right)!}\times {0.80}^{13}\times {0.20}^2\right)\\ +\left(\frac{15!}{14!\left(15-14\right)!}\times {0.80}^{14}\times {0.20}^1\right)+\left(\frac{15!}{15!\left(15-15\right)!}\times {0.80}^{15}\times {0.20}^0\right)\end{array}\right]\\ =\left(0.2501+0.2309+0.1319+0.0352\right)\end{array}$              

                $=0.648$

Therefore, the probability that “12 or more people will show significant improvement” is 0.648.