In an experiment, college students were given either four quarters or a $\$ 1$ bill and they could either keep the money or spend it on gum. The results are summarized in the table. Complete parts (a) through (c) below.
\begin{tabular}{c|c|c}
\hline & Purchased Gum & Kept the Money \\
\hline Students Given Four Quarters & 31 & 18 \\
\hline Students Given a \$1 Bill & 13 & 25 \\
\hline
\end{tabular}
a. Find the probability of randomly selecting a student who spent the money, given that the student was given four quarters.
The probability is (Round to three decimal places as needed.)
b. Find the probability of randomly selecting a student who spent the money, given that the student was given a $\$ 1$ bill.
The probability is
(Round to three decimal places as needed.)
Answer
Final Answer: The probability of randomly selecting a student who spent the money, given that the student was given four quarters is approximately \(\boxed{0.633}\). The probability of randomly selecting a student who spent the money, given that the student was given a $1 bill is approximately \(\boxed{0.342}\).
Step 1 ：Given the table of results from the experiment, we are asked to find the conditional probabilities of a student spending the money given that they were given either four quarters or a $1 bill.
Step 2 ：For part a, we need to find the probability that a student spent the money given that they were given four quarters. This is calculated by dividing the number of students who were given four quarters and spent the money by the total number of students who were given four quarters. From the table, we see that 31 students were given four quarters and spent the money, and a total of 49 students were given four quarters. Therefore, the probability is \(\frac{31}{49} \approx 0.633\).
Step 3 ：For part b, we need to find the probability that a student spent the money given that they were given a $1 bill. This is calculated by dividing the number of students who were given a $1 bill and spent the money by the total number of students who were given a $1 bill. From the table, we see that 13 students were given a $1 bill and spent the money, and a total of 38 students were given a $1 bill. Therefore, the probability is \(\frac{13}{38} \approx 0.342\).
Step 4 ：Final Answer: The probability of randomly selecting a student who spent the money, given that the student was given four quarters is approximately \(\boxed{0.633}\). The probability of randomly selecting a student who spent the money, given that the student was given a $1 bill is approximately \(\boxed{0.342}\).
