Question

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 four quarters.

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

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 given four quarters, is equal to 0.628.

b. The probability of selecting a studentwho spent the money, given that the student was given a $1 bill, is equal to 0.261.

c. Students who have four quarters have a greater tendency to spend the money than students who have a $1 bill.

Step-by-step solution

Given information

The number of students who were given a $1 bill and four quarters is tabulated.

They are further bifurcated into two categories: students who purchased gum and students who kept the money.

Define conditional probability

The conditional probability of an event is the probability of the occurrence of an event subject to a condition that another event has previously occurred. It has the following formula:

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

Compute the conditional probabilities

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:

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

a.

The total number of students is 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 who spent the money is 27.

The probability of selecting a student who was given 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 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 provided 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 given 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 equal to 0.261.

Interpret the results

c.

The probability of selecting a student who spent the money, given that the student was provided four quarters, is much greater than the probability of selecting a student who spent the money, given that the student was provided a $1 bill.

Thus, the students who received four quarters have a greater tendency to spend the money than students who received a $1 bill.

The results suggest that the students with four quarters have significantly higher chances of spending the money to purchase the gum.