Step 1
Given,
According to a study by Dr. John McDougall of his live-in weight loss program at St. Helena Hospital, the people who follow his program lose between 6 and 15 pounds a month until they approach trim body weight.
Let's suppose that the weight loss is uniformly distributed.
Step 2
The random variable \(\displaystyle{X}=\) Weight loss in pounds
Given \(\displaystyle{a}={6},{b}={15}\)

\(\displaystyle{X}\sim{U}{\left({a},{b}\right)}\Rightarrow{X}\sim{U}{\left({6},{15}\right)}\)

\(\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{b}-{a}}}\)

\(\displaystyle=\frac{{1}}{{{15}-{6}}}\)

\(\displaystyle=\frac{{1}}{{9}}\)

\(\displaystyle\mu=\frac{{{a}+{b}}}{{2}}\)

\(\displaystyle={\left({6}+\frac{{15}}{{2}}\right.}\)

\(\displaystyle=\frac{{21}}{{2}}\)

\(\displaystyle={10.5}\)

\(\displaystyle\sigma=\frac{{{b}-{a}}}{\sqrt{{{12}}}}\)

\(\displaystyle=\frac{{{15}-{6}}}{\sqrt{{{12}}}}\)

\(\displaystyle=\frac{{9}}{\sqrt{{{12}}}}\)

\(\displaystyle={2.5980}\) Step 3 The probability that the individual lost more than 8 pounds in a month: \(\displaystyle{P}{\left({x}{>}{8}\right)}=\frac{{{b}-{x}}}{{{b}-{a}}}\)

\(\displaystyle=\frac{{{15}-{8}}}{{{15}-{6}}}\)

\(\displaystyle={0.7778}\) (rounded off to 4 decimals) Step 4 Suppose it is known that the individual lost more than 9 pounds in a month. The probability that he lost less than 13 pounds in the month: \(\displaystyle{P}{\left({x}{<}{13}{m}{i}{d}{x}{>}{9}\right)}={P}\frac{{{9}{<}{x}{<}{13}}}{{P}}{\left({x}{>}{9}\right)}\)

\(\displaystyle{\frac{{{x}_{{{2}}}-{x}_{{{1}}}}}{{{b}-{a}}}}\ {\frac{{{b}-{x}}}{{{b}-{a}}}}\)

\(\displaystyle=\frac{{\frac{{{13}-{9}}}{{{15}-{6}}}}}{{\frac{{{15}-{9}}}{{{15}-{6}}}}}\)

\(\displaystyle=\frac{{\frac{{4}}{{9}}}}{{\frac{{6}}{{9}}}}\)

\(\displaystyle=\frac{{2}}{{3}}\)

\(\displaystyle={0.6667}\) (rounded off to 4 decimals)

\(\displaystyle{X}\sim{U}{\left({a},{b}\right)}\Rightarrow{X}\sim{U}{\left({6},{15}\right)}\)

\(\displaystyle{f{{\left({x}\right)}}}=\frac{{1}}{{{b}-{a}}}\)

\(\displaystyle=\frac{{1}}{{{15}-{6}}}\)

\(\displaystyle=\frac{{1}}{{9}}\)

\(\displaystyle\mu=\frac{{{a}+{b}}}{{2}}\)

\(\displaystyle={\left({6}+\frac{{15}}{{2}}\right.}\)

\(\displaystyle=\frac{{21}}{{2}}\)

\(\displaystyle={10.5}\)

\(\displaystyle\sigma=\frac{{{b}-{a}}}{\sqrt{{{12}}}}\)

\(\displaystyle=\frac{{{15}-{6}}}{\sqrt{{{12}}}}\)

\(\displaystyle=\frac{{9}}{\sqrt{{{12}}}}\)

\(\displaystyle={2.5980}\) Step 3 The probability that the individual lost more than 8 pounds in a month: \(\displaystyle{P}{\left({x}{>}{8}\right)}=\frac{{{b}-{x}}}{{{b}-{a}}}\)

\(\displaystyle=\frac{{{15}-{8}}}{{{15}-{6}}}\)

\(\displaystyle={0.7778}\) (rounded off to 4 decimals) Step 4 Suppose it is known that the individual lost more than 9 pounds in a month. The probability that he lost less than 13 pounds in the month: \(\displaystyle{P}{\left({x}{<}{13}{m}{i}{d}{x}{>}{9}\right)}={P}\frac{{{9}{<}{x}{<}{13}}}{{P}}{\left({x}{>}{9}\right)}\)

\(\displaystyle{\frac{{{x}_{{{2}}}-{x}_{{{1}}}}}{{{b}-{a}}}}\ {\frac{{{b}-{x}}}{{{b}-{a}}}}\)

\(\displaystyle=\frac{{\frac{{{13}-{9}}}{{{15}-{6}}}}}{{\frac{{{15}-{9}}}{{{15}-{6}}}}}\)

\(\displaystyle=\frac{{\frac{{4}}{{9}}}}{{\frac{{6}}{{9}}}}\)

\(\displaystyle=\frac{{2}}{{3}}\)

\(\displaystyle={0.6667}\) (rounded off to 4 decimals)