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Chapter 4: Problem 79

A 鈥渇riend鈥� offers you the following 鈥渄eal.鈥� For a \(\$ 10\) fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. However, each envelope contains a coupon for a free gift. 鈥� Ten of the coupons are for a free gift worth \(\$ 6\). 鈥� Eighty of the coupons are for a free gift worth \(\$ 8\). 鈥� Six of the coupons are for a free gift worth \(\$ 12\). 鈥� Four of the coupons are for a free gift worth \(\$ 40\). Based upon the financial gain or loss over the long run, should you play the game? a. Yes, I expect to come out ahead in money. b. No, I expect to come out behind in money. c. It doesn鈥檛 matter. I expect to break even.

Short Answer

Expert verified
No, you expect to come out behind in money.

Step by step solution

01

List Possible Outcomes

First, we identify the possible outcomes when selecting an envelope. There are 100 envelopes in total, with coupons valued at \\(6, \\)8, \\(12, and \\)40.
02

Calculate Probabilities

Next, we calculate the probability of selecting each coupon. For a gift worth \\(6, the probability is \(\frac{10}{100} = 0.1\). For \\)8, it is \(\frac{80}{100} = 0.8\). For \\(12, it is \(\frac{6}{100} = 0.06\). For \\)40, it is \(\frac{4}{100} = 0.04\).
03

Calculate Expected Value of Gift

We calculate the expected value (EV) by multiplying the value of each gift by its probability, then summing up these products: \(EV = (6 \times 0.1) + (8 \times 0.8) + (12 \times 0.06) + (40 \times 0.04)\).
04

Simplify the Expected Value Calculation

Perform the calculations: \(EV = 0.6 + 6.4 + 0.72 + 1.6 = 9.32\). So, the expected value of a gift in an envelope is \$9.32.
05

Determine Financial Outcome

Compare the expected value of the gift (\\(9.32) to the \\)10 fee for playing the game. Since \(9.32 < 10\), you are expected to have a loss of \$0.68 per game on average.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability
Understanding probability is crucial when analyzing scenarios with multiple potential outcomes. In this exercise, the probabilities of selecting different values from the envelopes were determined by the number of each type of coupon in proportion to the total number. There are 100 envelopes, and each one has a specific chance of being chosen.

Here's how you can think about it:
  • The probability of drawing a coupon worth \( \\(6 \) is calculated as the number of \( \\)6 \) coupons (10) divided by the total envelopes (100), which equals \( 0.1 \) or 10%.
  • Similarly, for \( \\(8 \) coupons, where 80 out of 100 envelopes include them, the probability is \( 0.8 \) or 80%.
  • For \( \\)12 \) coupons, with only 6 available, the probability becomes \( 0.06 \) or 6%.
  • Lastly, the chance of picking the rare \( \$40 \) coupon is a slim \( 0.04 \) or 4%.
These probabilities are fundamental building blocks for further statistical calculations, like finding the expected value.
Outcomes
Outcomes refer to the possible results of an event, in this case, picking an envelope at random. Each outcome has a specific associated value based on the contents of the envelope you might select.

- For this game, the possible financial outcomes are receiving a coupon worth \( \\(6, \\)8, \\(12, \) or \( \\)40 \). Due to the distribution of the coupons, the \( \$8 \) outcome is most likely to occur.

Understanding potential outcomes helps you grasp what could happen in any repeated trials, helping you prepare for the financial implications of your actions.
Financial Decision
Making a financial decision often involves comparing potential gains with associated costs. The decision here boils down to whether paying the \( \\(10 \) fee to pick an envelope is a wise choice.

To make informed decisions, one must weigh the expected financial result. By comparing the expected value of the outcomes (\( \\)9.32 \)) calculated from the probabilities against the cost of playing \( (\\(10) \), you can decide logically:

- Since 9.32 is less than 10, after considering the probabilities, playing results in a negative expected gain of \( -\\)0.68 \). Hence, the game isn鈥檛 financially beneficial over the long run.
Statistical Analysis
Statistical analysis helps to make sense of data and make predictions. In this scenario, we've performed a simplified form of statistical analysis by determining the expected value from the probability distribution of the gift coupons.

Expected value gives you a single number representing the average outcome over many trials, calculated via:
  • Calculating the expected value: Multiply each value by its probability and sum them up: \[(6 \times 0.1) + (8 \times 0.8) + (12 \times 0.06) + (40 \times 0.04) = 9.32\]
  • This means, on average, you'd expect to value each envelope at \( \$9.32 \).
This approach helps businesses and individuals to weigh potential outcomes against costs, enabling more informed decision-making. Statistical thinking lays the foundation for unbiased, data-driven financial choices.

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Most popular questions from this chapter

Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bowl. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ____ (__,___) d. How many adults in America do you expect to survey until you find one who will watch the Super Bowl? e. Find the probability that you must ask seven people. f. Find the probability that you must ask three or four people.

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. In words, define the random variable \(X.\)

In one of its Spring catalogs, L.L. Bean庐 advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. Each page may be picked at most once.. a. In words, define the random variable X. b. List the values that X may take on. C. Give the distribution of \(X . X \sim\) ____ (___,____) d. How many pages do you expect to advertise footwear on them? e. Calculate the standard deviation.

Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and eight who are not. We are interested in the number on the committee who are not technically proficient. a. In words, define the random variable X. b. List the values that X may take on. C. Give the distribution of \(X . X \sim\) ____ (___,____) d. How many instructors do you expect on the committee who are not technically proficient? e. Find the probability that at least five on the committee are not technically proficient. f. Find the probability that at most three on the committee are not technically proficient.

Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing post-graduate research. He has the following probability distribution. $$\begin{array}{|c|c|}\hline x & {P(x)} & {x^{\star} P(x)} \\ \hline 1 & {0.35} \\ \hline 2 & {0.20} \\ \hline 3 & {0.15} \\ \hline 4 & {} \\\ \hline 5 & {0.10} \\ \hline 6 & {0.05} \\ \hline\end{array}$$ Define the random variable \(X\).
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