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	16	35

a. find the probability of randomly selecting a student who spent the money, given that the student was given four quarters.
the probability is 0.641 (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 0.359 (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 than a student given a $1 bill.
b. a student given four quarters is more likely to have spent the money.
c. a student given four quarters is more likely to have kept the money than a student given a $1 bill.
d. a student given four quarters is more likely to have kept the money.

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$. In terms of frequency, if we want to find the probability of 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 part (a)

For part (a), the number of students given four - quarters is $n = 25 + 14=39$. The number of students who spent the money (purchased gum) among those given four - quarters is $n_1 = 25$. So the probability $P=\frac{25}{39}\approx0.641$.

Step3: Calculate part (b)

For part (b), the number of students given four - quarters is still $n = 25+14 = 39$. The number of students who kept the money among those given four - quarters is $n_2 = 14$. So the probability $P=\frac{14}{39}\approx0.359$.

Step4: Analyze part (c)

The probability that a student given four - quarters spent the money is approximately $0.641$, and the probability that a student given a $\$1$ bill spent the money is $\frac{16}{16 + 35}=\frac{16}{51}\approx0.314$. Since $0.641>0.314$, a student given four - quarters is more likely to have spent the money.

Answer:

a. $0.641$
b. $0.359$
c. B. A student given four quarters is more likely to have spent the money