Question

Denomination Effect. In Exercises 13–16, use the data in the following table. In an experiment to study the effects of using four quarters or a \(1 bill, college students were given either four quarters or a \)1 bill and they could either keep the money or spend it on gum. The results are summarized in the table (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 Four Quarters	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 four quarters.

b. Find the probability of randomly selecting a student who kept the money, given that the student was given four quarters.

c. What do the preceding results suggest?

Short answer

a. The probability of selecting a student who spent the money, given that the student was provided four quarters, is equal to 0.628.

b. The probability of selecting a studentwho kept the money, given that the student was provided four quarters, is equal to 0.372.

c. If students are given four quarters, they tend to spend the money rather than keeping it.

Step-by-step solution

Given information

The number of students who were given four quarters as well as a $1 bill is tabulated based on whether they kept the money or spent it.

Conditional probability

As the name suggests, the conditional probability of an eventis the probability of the event happening subject to a condition that an event has already occurred. It has the following formula:

Compute the conditional probability

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 (purchased gum).

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

The following table shows the necessary totals:

	Purchased Gum	Kept the Money	Totals
Students Given Four Quarters	27	16	43
Students Given a $1 bill	12	34	46
Totals	39	50	89

a.

The total number of students is equal to 89.

The number of students who were given four quarters is equal to 43.

The probability of selecting a student who was provided four quarters is given by:

$P\left(A\right)=\frac{43}{89}$

The number of students who were given four quarters and spent the money is equal to 27.

The probability of selecting a student who was provided four quarters and spent the money is given by:

$P\left(AandC\right)=\frac{27}{89}$

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

$$\begin{gathered} P\left(C|A\right)=\frac{P\left(AandC\right)}{P\left(A\right)} \\ =\frac{\frac{27}{89}}{\frac{43}{89}} \\ =\frac{27}{43} \\ =0.628 \end{gathered}$$

Therefore, the probability of selecting a student who spent the money, given that he/she was provided four quarters, is equal to 0.628.

b.

The number of students who were given four quarters and kept the money is equal to 16.

The probability of selecting a student who was given four quarters and kept the money is given by:

$P\left(AandD\right)=\frac{16}{89}$

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

$$\begin{gathered} P\left(D|A\right)=\frac{P\left(AandD\right)}{P\left(A\right)} \\ =\frac{\frac{16}{89}}{\frac{43}{89}} \\ =\frac{16}{43} \\ =0.372 \end{gathered}$$

Therefore, the probability of selecting a student who kept the money, given that he/she was provided four quarters, is equal to 0.372.

 Step 4: Interpret the results

c.

As the probability of selecting a student who spent the money is greater than that of a student who kept the money when offered with four quarters, it can be concluded that students tend to spend money rather than keep it.

Thus, it is suggestive that there is a significantly higher tendency to purchase gum rather than keep the money if offered four quarters.