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 kept 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 a $1 bill.

c. What do the preceding results suggest?

Short answer

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

b. The probability of selecting a studentwho kept the money, given that the student was provided a $1 bill, is equal to 0.739

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

Step-by-step solution

Given information

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

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

Define conditional probability

The conditional probability of an eventis computed with reference to a prior event that occurred in the past. 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 43.

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

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

The number of students who were given four quarters and kept the money is 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 given 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 given four quarters, is equal to 0.372.

b.

The total number of students is 89.

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

The probability of selecting a student who was given 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 given 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 given a $1 bill, is equal to 0.739.

Interpret the results

c.

The probability of selecting a student who kept the money, given that the student was given a $1 bill, is 0.739. It is much greater than the probability of selecting a student who kept the money when provided with four quarters (0.372).

Thus, the students who received a $1 bill have a greater tendency of keeping the money as compared to the students who received four quarters.

The results suggest higher chances of a student keeping the money when given a $1 bill.