Question

You bet! You roll a die. If it comes up a 6 , you win \(\$ 100\). If not, you get to roll again. If you get a 6 the second time, you win \(\$ 50\). If not, you lose. a) Create a probability model for the amount you win. b) Find the expected amount you'll win. c) What would you be willing to pay to play this game?

Short answer

Expected win: \(\$23.61\). Pay no more than \(\$23.61\) to play.

Step-by-step solution

Understand the Possibilities

This game has three possible outcomes: you win \(\\(100\) if you roll a 6 on the first roll, you win \(\\)50\) if you roll a 6 on the second roll after not getting a 6 on the first roll, and you win \(\$0\) if you don't roll a 6 on either roll.

Calculate Probabilities for Each Outcome

The probability of rolling a 6 on a six-sided die is \(\frac{1}{6}\). The probability of not rolling a 6 is \(\frac{5}{6}\). For the outcome of winning \(\\(100\), you need to roll a 6 on the first try, so the probability is \(\frac{1}{6}\). For winning \(\\)50\), you must not get a 6 on the first roll (\(\frac{5}{6}\)) and then roll a 6 on the second (\(\frac{1}{6}\)), which results in a combined probability of \(\frac{5}{6} \times \frac{1}{6} = \frac{5}{36}\). For winning \(\$0\), the probability is simply failing to roll a 6 in both trials, which is \(\frac{5}{6} \times \frac{5}{6} = \frac{25}{36}\).

Create the Probability Model

Your probability model, which pairs outcomes with their probabilities, is:- Win \(\\(100\): Probability = \(\frac{1}{6}\)- Win \(\\)50\): Probability = \(\frac{5}{36}\)- Win \(\$0\): Probability = \(\frac{25}{36}\).

Calculate Expected Value

The expected value is calculated by multiplying each outcome by its probability and summing the results: \(E(X) = 100 \times \frac{1}{6} + 50 \times \frac{5}{36} + 0 \times \frac{25}{36}\). This simplifies to:\(E(X) = 16.67 + 6.94 + 0 = 23.61\).Therefore, the expected amount you win is \(\$23.61\).

Determine the Payment for Playing

To determine what you should be willing to pay to play the game, consider your expected earnings. Rationally, it's reasonable to pay up to the expected value of \(\$23.61\) to play.

Key concepts

Expected Value

The concept of expected value is key in determining what you should reasonably expect to win or lose over many iterations of a particular game or event. In this example with the dice game, you want to estimate how much you could win on average if you played the game many times.

To find the expected value, you multiply each possible outcome by its probability, then add them all together. For this dice game:

• If you roll a 6 on the first try: You win \\(100. The probability of this is \( \frac{1}{6} \).
• If you roll a 6 on the second try: You win \\)50. First, you didn’t roll a 6 (\( \frac{5}{6} \)), then you rolled a 6 on the second chance (\( \frac{1}{6} \)), making the combined probability \( \frac{5}{36} \).
• If you don’t roll a 6 at all, you win \\(0 with probability \( \frac{25}{36} \).

Add up the weighted outcomes:\[ E(X) = 100 \times \frac{1}{6} + 50 \times \frac{5}{36} + 0 \times \frac{25}{36}\]This results in an expected value of \\)23.61, indicating that over time, you can expect to win about \$23.61 per game. This serves as a rational guide to decide how much you might be willing to pay to play.

Game Theory

Game theory is all about making decisions strategically in scenarios where the outcome is affected by the actions of other "players." In this dice game, the strategies aren't just about competition but deciding whether the game is worth playing based on possible outcomes and probabilities.

The key idea here is to evaluate the "fairness" of the game. A fair game in game theory would be one where the expected gains and losses balance out. Thus, the cost to play should not exceed the expected value of \\(23.61. If it does, you are more likely to lose money over time.

Applying game theory, you look for ways to maximize your advantage, ensuring that when the game is played repeatedly, you come out on top. Understanding this balance helps you decide whether to play and how much risk you are willing to take. Always consider other factors that could influence your decision, such as how much value you personally place on the chance of winning more than \\)23.61.

Probability Calculation

Probability calculation involves figuring out how likely different outcomes are and is essential to making informed predictions about the game. This dice game needed precise probability calculation to model the chances of each win or loss.

Here's a quick breakdown of how you calculate these probabilities for the game scenarios:

• Win \\(100: Only if a 6 is rolled on the first attempt. Makes probability \( \frac{1}{6} \).
• Win \\)50: If the first roll is not a 6 (\( \frac{5}{6} \)) and the second roll is a 6 (\( \frac{1}{6} \)), the probability of this compound event is \( \frac{5}{6} \times \frac{1}{6} = \frac{5}{36} \).
• Win \$0: Failing to roll a 6 in both rolls gives a probability of \( \frac{5}{6} \times \frac{5}{6} = \frac{25}{36} \).

Probability is about calculating these ratios and understanding what they mean in a real-world context. It helps visualize and comprehend the chances of various outcomes happening, allowing for smarter, data-driven decisions in games and beyond.