Contingency Tables and Probability Trees · Try It 3

Fred's preference to drink coffee or iced tea depends on the season. When it is summer, Fred's preference to drink coffee is $\frac{1}{4},$ and his preference to drink iced tea is $\frac{3}{4}.$When it is rainy season, Fred's preference to drink coffee is $\frac{1}{2},$and his preference to drink iced tea is $\frac{1}{2}.$ When it is winter, Fred's preference to drink coffee is $\frac{1}{5},$ and his preference to drink iced tea is $\frac{1}{5}.$ A day randomly selected in a year has a probability of $\frac{1}{3}$ for each season. The event here is selection of a day randomly in a year.

\begin{array}{|c|c|c|c|c|}\hline \text{Season} & \text{Summer} & \text{Rainy} & \text{Winter} & \text{Total} \\ \hline \text{Coffee} & \frac{1}{12} & \frac{1}{6} & \frac{4}{15} & \text{__} \\ \hline \text{Iced tea} & \frac{1}{4} & \frac{1}{6} & \frac{1}{15} & \text{__} \\ \hline \text{Total} & \text{__} & \text{__} & \text{__} & \text{1} \\ \hline\end{array}

• The first entry is $\left(\frac{1}{4}\right)\left(\frac{1}{3}\right)=\frac{1}{12}.$
• The second entry is $\left(\frac{3}{4}\right)\left(\frac{1}{3}\right)=\frac{1}{4}.$
Verify the remaining entries.

(a) Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is 1.
(b) What is the probability that Fred will drink iced tea?
(c) What is the probability that the day is in summer or rainy season given that Fred drinks iced tea?

OpenStax, Introductory Business Statistics 2e · §3.4 Contingency Tables and Probability Trees · Try It 3