Question

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped $64$stamped, addressed envelopes with $\$10$cash in different classrooms on the George Washington campus. $44\%$were returned overall. From the economics classes $56\%$of the envelopes were returned. From the business, psychology, and history classes $31\%$were returned.

Let:R = money returned;E = economics classes; O = other classes

a. Write a probability statement for the overall percent of money returned.

b. Write a probability statement for the percent of money returned out of the economics classes.

c. Write a probability statement for the percent of money returned out of the other classes.

d. Is money being returned independent of the class? Justify your answer numerically and explain it.

e. Based upon this study, do you think that economists are more selfish than other people? Explain why or why not. Include numbers to justify your answer.

Short answer

(a) The probability for the overall percentage of money returned is $0.44$.

(b) The probability for the percent of money returned out of the economic classes is$0.56$

(c) The probability for the percent of money returned out of the other classes is $0.31$.

(d) The money being returned is not independent of the class.

(e) Based upon this study, we cannot say that economists are more selfish than other people.

Step-by-step solution

Given information (part a)

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped $64$stamps, addressed envelopes with $\$10$cash in different classrooms on the George Washington campus. $44\%$were returned overall. From the economic classes, $56\%$of the envelopes were returned. From the business, psychology, and history classes $31\%$were returned.

Explanation (part a)

Let $\text{R}=$money returned

$\text{E}=$economic classes

$\text{O}=$other classes

Given that $44\%$were returned overall.

Thus, the probability for the overall percent of money returned is given as

$P\left(R\right)=0.44$

Given information (part b) 

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped $64$stamps, addressed envelopes with $\$10$cash in different classrooms on the George Washington campus. $44\%$were returned overall. From the economic classes, $56\%$ of the envelopes were returned. From the business, psychology, and history classes $31\%$ were returned.

Explanation (part b)

Let $\text{R}=$money returned

$\text{E}=$economic classes

$\text{O}=$other classes

Given that from the economic classes $56\%$of the envelopes were returned.

Thus, the probability for the percent of money returned out of the economic classes is given as$P\left(R|E\right)=0.56$

Given information (part c)

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped $64$stamps, addressed envelopes with $\$10$cash in different classrooms on the George Washington campus. $44\%$were returned overall. From the economic classes, $56\%$ of the envelopes were returned. From the business, psychology, and history classes $31\%$ were returned.

Explanation (part c)

Let $\text{R}=$money returned

$\text{E}=$economic classes

$\text{O}=$other classes

Given that from the other classes (business, psychology, and history classes), $31\%$were returned.

Thus, the probability for the percent of money returned out of the other classes is given as

$P\left(R|O\right)=0.31$

Given information (part d)

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped $64$stamps, addressed envelopes with $\$10$cash in different classrooms on the George Washington campus. $44\%$were returned overall. From the economic classes, $56\%$ of the envelopes were returned. From the business, psychology, and history classes $31\%$ were returned.

Explanation (part d)

Let the events be

$\text{R}=$money returned

$\text{E}=$economic classes

$\text{O}=$other classes

Here, Let us first check if $P\left(R|E\right)=P\left(R\right)$

We have

$$\begin{gathered} P\left(R|E\right)=0.56 \\ P\left(R\right)=0.44 \\ P(R\mid E)\ne P\left(R\right) \end{gathered}$$

The first condition is not satisfied.

Therefore, the events are not independent.

Given information (part e)

Three professors at George Washington University did an experiment to determine if economists are more selfish than other people. They dropped $64$stamps, addressed envelopes with $\$10$cash in different classrooms on the George Washington campus. $44\%$were returned overall. From the economic classes, $56\%$of the envelopes were returned. From the business, psychology, and history classes $31\%$were returned.

Explanation (part e)

Let the events be

$\text{R}=$money returned

$\text{E}=$economic classes

$\text{O}=$other classes

Here, Let us first check if $P\left(R|E\right)=P\left(R\right)$

We have

$$\begin{gathered} P\left(R|E\right)=0.56 \\ P\left(R\right)=0.44 \\ \text{Thus},P\left(R|E\right)>P\left(R\right) \end{gathered}$$