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Chapter 5: Problem 84

Consider the following two lottery-type games: Game 1 : You pick one number between 1 and \(50 .\) After you have made your choice, a number between 1 and 50 is selected at random. If the selected number matches the number you picked, you win. Game 2 : You pick two numbers between 1 and \(10 .\) After you have made your choices, two different numbers between 1 and 10 are selected at random. If the selected numbers match the two you picked, you win. a. The cost to play either game is \(\$ 1,\) and if you win you will be paid \(\$ 20 .\) If you can only play one of these games, which game would you pick and why? Use relevant probabilities to justify your choice. b. For either of these games, if you plan to play the game 100 times, would you expect to win money or lose money overall? Explain.

Short Answer

Expert verified
We would choose to play Game 1 because it has a higher probability of winning (\(P(Game\,1\,Win) = \frac{1}{50}\) or 0.02) compared to the probability of winning Game 2 (\(P(Game\,2\,Win) = \frac{1}{90}\) or 0.0111). However, the expected value for both games is negative, with Game 1 at -0.6 and Game 2 at -0.7778. This means we would expect to lose money overall if we played either game 100 times.

Step by step solution

01

Calculate Game 1's Probability of Winning

To calculate the probability of winning Game 1, we'll use the total possible outcomes and the number of favorable outcomes. We have 1 favorable outcome (matching the random number) out of 50 possible outcomes (since there are 50 available numbers): \(P(Game\,1\,Win) = \frac{1}{50}\)
02

Calculate Game 2's Probability of Winning

In Game 2, we need to find the probability of correctly picking two different numbers between 1 and 10. The total number of possible outcomes is: \(10 \times 9 = 90\) (because we can't pick the same number twice) The number of favorable outcomes is 1 (matching our two selected numbers). Thus, the probability of winning Game 2 is: \(P(Game\,2\,Win) = \frac{1}{90}\)
03

Compare Game 1 and Game 2 Probabilities

Now that we have the probabilities of winning each game, we can compare them. The probability of winning Game 1 is \(1/50 \approx 0.02\), whereas the probability of winning Game 2 is \(1/90 \approx 0.0111\). Since the probability of winning Game 1 is higher, we would choose to play Game 1.
04

Calculate Expected Gains/Losses for 100 Games

Now let's calculate whether we'd expect to gain or lose money if we played each game 100 times. We know that the cost of playing each game is \(1, so our total expenses after 100 games will be \)100. If we win the game, we get a reward of $20. We would calculate the expected value by multiplying the probability of winning by the winnings and subtracting the cost of playing each game. For Game 1: Expected Value = (Probability of Winning x Winnings) - Cost of Playing = \((1/50 \times 20) - 1 = 0.4 - 1 = -0.6\) For Game 2: Expected Value = (Probability of Winning x Winnings) - Cost of Playing = \((1/90 \times 20) - 1 = 0.2222 - 1 = -0.7778\) In both cases, the expected values are negative, which means that after playing 100 games, we would expect to lose money overall. In Game 1, we would lose \(0.6 per game, and in Game 2, we would lose \)0.7778 per game on average if we played 100 times. So, if we have to choose one game to play, we would select Game 1 because it has a higher probability of winning. However, we can see that based on the expected values for both games, we would expect to lose money overall if we played either of them 100 times.

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Key Concepts

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

Expected Value in Gaming
When choosing a game of chance, it's essential to understand the expected value, which represents the average amount one can expect to win or lose per bet in the long run. Calculating the expected value allows players to assess the fairness or potential profitability of a game.

For instance, in the textbook exercise, we consider the two lottery-type games. The expected value for each game is determined by multiplying the probability of winning by the payout (20), then subtracting the cost to play (1). Since both games have a negative expected value, players are likely to lose money over time. In Game 1, players lose an average of 0.6 per play, while in Game 2, the average loss is 0.7778 per play. Making an informed decision involves choosing the game with the higher expected value, even if it's still a net loss, because it minimizes potential losses over time.
Probability Calculations in Games
Probability calculations are the backbone of understanding games of chance. These calculations help players determine the likelihood of winning and are expressed as a fraction or a percentage.

In our exercise's context, we use basic combinatorial principles to calculate the probability of winning. For Game 1, the probability is simply 1 out of 50, or 2%. In Game 2, since two distinct numbers are chosen, we calculate the odds by recognizing there are 90 distinct possible pairs when choosing two numbers from 1 to 10. However, only one of those pairs will be the winning combination, leading to a probability of 1 in 90, or about 1.11%. Correctly understanding this process is crucial for assessing the odds and making strategic decisions when faced with multiple game options.
Comparing Probabilities to Make Decisions
Comparing probabilities is a key decision-making tool in games. When faced with multiple options, like the games in the exercise, players should weigh the probabilities to identify which game offers a better chance of winning.

In the exercise, we compared the probability of winning Game 1 (1 in 50) with that of Game 2 (1 in 90). Upon comparison, it's clear that Game 1 provides a higher chance of success. Despite both outcomes being negative when considering the expected values, the comparison leads to the rational choice of opting for Game 1. Engaging in comparative analysis of probabilities can help in choosing the less risky option, even when neither guarantees a win, thus exposing the player to less potential financial loss over time.

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Most popular questions from this chapter

The report "Twitter in Higher Education: Usage Habits and Trends of Today's College Faculty" describes a survey of nearly 2000 college faculty. The report indicates the following: \(30.7 \%\) reported that they use Twitter, and \(69.3 \%\) said that they do not use Twitter. Of those who use Twitter, \(39.9 \%\) said they sometimes use Twitter to communicate with students. Of those who use Twitter, \(27.5 \%\) said that they sometimes use Twitter as a learning tool in the classroom. Consider the chance experiment that selects one of the study participants at random. a. Two of the percentages given in the problem specify unconditional probabilities, and the other two percentages specify conditional probabilities. Which are conditional probabilities? How can you tell? b. Suppose the following events are defined: \(T=\) event that selected faculty member uses Twitter \(C=\) event that selected faculty member sometimes uses Twitter to communicate with students \(L=\) event that selected faculty member sometimes uses Twitter as a learning tool in the classroom Use the given information to determine the following probabilities: i. \(P(T)\) iii. \(P(C \mid T)\) ii. \(P\left(T^{C}\right)\) iv. \(\quad P(L \mid T)\) c. Construct a hypothetical 1000 table using the given probabilities and use the information in the table to calculate \(P(C),\) the probability that the selected study participant sometimes uses Twitter to communicate with students. d. Construct a hypothetical 1000 table using the given probabilities and use the information in the table to calculate the probability that the selected study participant sometimes uses Twitter as a learning tool in the classroom.

For a monthly subscription fee, a video download site allows people to download and watch up to five movies per month. Based on past download histories, the following table gives the estimated probabilities that a randomly selected subscriber will download 0,1,2,3,4 or 5 movies in a particular month. $$ \begin{array}{|lcccccc|} \hline \text { Number of downloads } & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Estimated probability } & 0.03 & 0.45 & 0.25 & 0.10 & 0.10 & 0.07 \\ \hline \end{array} $$ If a subscriber is selected at random, what is the estimated probability that this subscriber downloads a. three or fewer movies? b. at most three movies? c. four or more movies? d. zero or one movie? e. more than one movie?

If you were to roll a fair die 1000 times, about how many sixes do you think you would observe? What is the probability of observing a six when a fair die is rolled?

Each time a class meets, the professor selects one student at random to explain the solution to a homework problem. There are 40 students in the class, and no one ever misses class. Luke is one of these students. What is the probability that Luke is selected both of the next two times that the class meets?

In a January 2016 Harris Poll, each of 2252 American adults was asked the following question: "If you had to choose, which ONE of the following sports would you say is your favorite?" ("Pro Football is Still America's Favorite Sport," www.theharrispoll.com/sports/Americas_Fav_Sport_2016. html, retrieved April 25,2017 ). Of the survey participants, \(33 \%\) chose pro football as their favorite sport. The report also included the following statement, "Adults with household incomes of \(\$ 75,000-<\$ 100,000(48 \%)\) are especially likely to name pro football as their favorite sport, while love of this particular game is especially low among those in \(\$ 100,000+$$ households \)(21 \%)\( Suppose that the percentages from this poll are representative of American adults in general. Consider the following events: \)F=\( event that a randomly selected American adult names pro football as his or her favorite sport \)L=\( event that a randomly selected American has a household income of \)\$ 75,000-<\$ 100,000\( \)H=\( event that a randomly selected American has a household income of \)\$ 100,000+$$ b. Are the events \(F\) and \(L\) mutually exclusive? Justify your answer. c. Are the events \(H\) and \(L\) mutually exclusive? Justify your answer. d. Are the events \(F\) and \(H\) independent? Justify your answer.
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