Problem 4 - Algebra and Trigonometry

Question

\(3-12\) . Find the expected value (or expectation) of the games described. Jane wins \(\$ 10\) if a die roll shows a six, and she loses \(\$ 1\) otherwise.

Step-by-Step Solution

Answer

Jane's expected value for each game is \(\frac{5}{6}\) dollars.

Step 1: Define Possible Outcomes

There are two possible outcomes when the die is rolled: either a six is rolled, or any other number is rolled (1 through 5).

Step 2: Identify Probabilities

The probability of rolling a six with a fair die is \( \frac{1}{6} \). The probability of not rolling a six (i.e., rolling a 1, 2, 3, 4, or 5) is \( \frac{5}{6} \).

Step 3: Determine Outcomes' Values

If a six is rolled, Jane wins \\(10. If any other number is rolled, Jane loses \\)1. These monetary outcomes form the event values.

Step 4: Calculate Expectation

The expected value (EV) is calculated by multiplying each outcome by its probability and summing the results. \[EV = (10 \times \frac{1}{6}) + (-1 \times \frac{5}{6})\] Evaluating this expression yields:\[EV = \frac{10}{6} + \frac{-5}{6} = \frac{5}{6}\]

Step 5: Interpret the Result

The expected value of \(\frac{5}{6}\) means that, on average, Jane can expect to win \(\frac{5}{6}\) of a dollar per game.

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. It ranges from 0 to 1, where 0 means the event will not happen, and 1 means it will certainly happen. In the context of Jane's game with the die, we have two events: rolling a six and not rolling a six.

• The probability of rolling a six with a fair die is \( \frac{1}{6} \).
• The probability of rolling any other number (1 through 5) is \( \frac{5}{6} \).

These probabilities help us in calculating the expected value by showing how often we expect each outcome to occur over repeated trials.

Outcomes

Each event that can occur when rolling a die is known as an outcome. A standard six-sided die has the following possible outcomes when rolled: 1, 2, 3, 4, 5, and 6. In Jane's game, however, these outcomes are categorized into two main groups based on the game's rules:

• Rolling a six, which results in winning $10.
• Rolling any of the numbers from 1 to 5, which results in losing $1.

Understanding these outcomes is crucial as they are employed in determining the expected value by associating each with its respective gain or loss.

Fair Die

A fair die is a die that has an equal probability of landing on any of its six faces when rolled. This is an important assumption, as it guarantees that each number from 1 to 6 is equally likely to occur. In Jane's game context:

• The probability of each side coming up is \( \frac{1}{6} \), making it a fair die.
• Each roll is independent of the previous one, noting the fairness in multiple game rounds.

This fairness is what allows us to use probabilities succinctly in calculating expected outcomes.

Game Theory

Game theory is the study of strategic decision making and can be applied to situations like Jane's game with the die. It involves analyzing different strategies to maximize a player's advantage, often through expectation values. In the context of Jane’s die game:

• The decision revolves around playing the game multiple times and aiming for a positive expected value.
• The expected value calculation shows Jane can expect, on average, to gain \( \frac{5}{6} \) of a dollar per game if she plays repeatedly.

Understanding these concepts gives players insights into whether engaging in such games could be beneficial over time.