Denomination Effect. In Exercises 13–16, use the data in the following table. In an experiment to study the effects of using a \(1 bill or a \)1 bill, college students were given either a \(1 bill or a \)1 bill and they could either keep the money or spend it on gum. The results are summarized in the table (based on data from “The Denomination Effect,” by Priya Raghubir and Joydeep Srivastava, Journal of Consumer Research, Vol. 36).

	Purchased Gum	Kept the Money
Students Given A \(1 bill	27	46
Students Given a \)1 bill	12	34

Denomination Effect

a. Find the probability of randomly selecting a student who spent the money, given that the student was given a \(1 bill.

b. Find the probability of randomly selecting a student who kept the money, given that the student was given a \)1 bill.

c. What do the preceding results suggest?

Some students were given a $1 bill or four quarters. The frequencies are categorized into two categories comprising whether they spent the money or not.

As the name suggests,theconditional probability of an eventis the probability of the occurrence of the event with reference to a prior condition. It has the following formula:

$P\left(B|A\right)=\frac{P\left(AandB\right)}{P\left(A\right)}$

Let A be the event of selecting a student who was given four quarters.

Let B be the event of selecting a student who was given a $1 bill.

Let C be the event of selecting a student who spent the money.

Let D be the event of selecting a student who kept the money.

The following table shows the necessary totals:

a.

The total number of students is 89.

The number of students who were given a $1 bill is equal to 46.

The probability of selecting a student who was given a $1 bill is given by:

$P\left(B\right)=\frac{46}{89}$

The number of students who were given a $1 bill and spent the money is 12.

The probability of selecting a student who was provided a $1 bill and spent the money is given by:

$P\left(BandC\right)=\frac{12}{89}$

The probability of selecting a student who spent the money, given that he/she was provided a $1 bill, is computed as follows:

$$\begin{gathered} P\left(C|B\right)=\frac{P\left(BandC\right)}{P\left(B\right)} \\ =\frac{\frac{12}{89}}{\frac{46}{89}} \\ =\frac{12}{46} \\ =0.261 \end{gathered}$$

Therefore, the probability of selecting a student who spent the money, given that he/she was provided a $1 bill, is 0.261.

b.

The number of students who were given a $1 bill and kept the money is 34.

The probability of selecting a student who was provided a $1 bill and kept the money is given by:

$P\left(BandD\right)=\frac{34}{89}$

The probability of selecting a student who kept the money, given that he/she was provided a $1 bill, is computed as follows:

$$\begin{gathered} P\left(D|B\right)=\frac{P\left(BandD\right)}{P\left(B\right)} \\ =\frac{\frac{34}{89}}{\frac{46}{89}} \\ =\frac{34}{46} \\ =0.739 \end{gathered}$$

Therefore, the probability of selecting a student who kept the money, given that he/she was provided a $1 bill, is equal to 0.739.

c.

The probability of selecting a student who kept the money is greater than that of a student who spent it (purchased gum), given that the student had a $1 bill.

Thus, the result suggests that it can be concluded that students tend to keep the money than spending it if they are given a $1 bill.