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.

students given four quarters	purchased gum	kept the money
students given a $1 bill	16	26

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 kept the money than a student given a $1 bill.
b. a student given four quarters is more likely to have spent 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.

Explanation:

Step1: Define the formula for conditional probability

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In the context of frequency - based data, if we want to find the probability of event $A$ (spent the money) given event $B$ (given four quarters), we use $P(A|B)=\frac{n(A\cap B)}{n(B)}$, where $n(A\cap B)$ is the number of students who both spent the money and were given four quarters, and $n(B)$ is the number of students who were given four quarters.

Step2: Calculate the probability of spending money given four quarters

The number of students given four quarters is $n = 31 + 19=50$. The number of students who were given four quarters and spent the money (purchased gum) is $31$. So the probability of randomly selecting a student who spent the money, given that the student was given four quarters is $P=\frac{31}{50}=0.62$.

Step3: Calculate the probability of keeping money given four quarters

The number of students who were given four quarters and kept the money is $19$. Using the same total number of students given four quarters ($n = 50$), the probability of randomly selecting a student who kept the money, given that the student was given four quarters is $P=\frac{19}{50}=0.38$.

Step4: Analyze the results

Since $0.62>0.38$, a student given four quarters is more likely to have spent the money.

Answer:

a. $0.62$
b. $0.38$
c. D. A student given four quarters is more likely to have spent the money.