Question

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)

	purchased gum	kept the money
students given a $1 bill	13	31

a. find the probability of randomly selecting a student who spent the money, given that the student was given four quarters.
the probability is (square) (round to three decimal places as needed.)
b. find the probability of randomly selecting a student who kept the money, given that the student was given four quarters.
the probability is (square) (round to three decimal places as needed.)
c. what do the preceding results suggest?
a. a student given four quarters is more likely to have spent the money.
b. a student given four quarters is more likely to have kept the money than a student given a $1 bill.
c. a student given four quarters is more likely to have kept the money.
d. a student given four quarters is more likely to have spent the money than a student given a $1 bill.

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In the context of frequency - tables, if we want to find the probability of an event $A$ given event $B$, we use $P(A|B)=\frac{n(A\cap B)}{n(B)}$, where $n(A\cap B)$ is the number of elements in the intersection of $A$ and $B$, and $n(B)$ is the number of elements in $B$.

Step2: Calculate probability for part a

The number of students given four quarters is $n(\text{four quarters})=25 + 19=44$. The number of students who spent the money and were given four quarters is $n(\text{spent and four quarters}) = 25$. So the probability of randomly selecting a student who spent the money, given that the student was given four quarters is $P(\text{spent}|\text{four quarters})=\frac{25}{44}\approx0.568$.

Step3: Calculate probability for part b

The number of students who kept the money and were given four quarters is $n(\text{kept and four quarters}) = 19$. So the probability of randomly selecting a student who kept the money, given that the student was given four quarters is $P(\text{kept}|\text{four quarters})=\frac{19}{44}\approx0.432$.

Step4: Analyze the results for part c

The probability that a student given four quarters spent the money ($0.568$) is higher than the probability that a student given a $\$1$ bill spent the money. The probability that a student given four quarters kept the money is $0.432$. A student given four quarters is more likely to have spent the money.

Answer:

a. $0.568$
b. $0.432$
c. A. A student given four quarters is more likely to have spent the money