Problem 20 - Algebra and Trigonometry

Question

A Game of Chance A bag contains two silver dollars and six slugs. A game consists of reaching into the bag and drawing a coin, which you get to keep. Determine the "fair price" of playing this game, that is, the price at which the player can be expected to break even if he or she plays the game many times (in other words, the price at which the player's expectation is zero).

Step-by-Step Solution

Answer

The fair price is $0.25 per game.

Step 1: Define the Probability of Each Outcome

There are a total of 8 coins in the bag: 2 silver dollars and 6 slugs. The probability of drawing a silver dollar is therefore \( \frac{2}{8} = \frac{1}{4} \) and the probability of drawing a slug is \( \frac{6}{8} = \frac{3}{4} \).

Step 2: Determine the Value of Each Outcome

In this game, drawing a silver dollar means you win $1, while drawing a slug means you win nothing ($0).

Step 3: Calculate the Expected Value

The expected value (E) is found by multiplying the value of each outcome by its probability and summing the results: \( E = (\text{Probability of Silver Dollar} \times \text{Value of Silver Dollar}) + (\text{Probability of Slug} \times \text{Value of Slug}) \).

Step 4: Plug in the Numbers

Substituting the known values: \( E = \left( \frac{1}{4} \times 1 \right) + \left( \frac{3}{4} \times 0 \right) = \frac{1}{4} \).

Step 5: Determine the Fair Price

The fair price of playing this game is the expected value per play, which is \( \frac{1}{4} \) or $0.25. This is the price at which the player's net gain over many plays is zero.

Key Concepts

Expected Value

In probability, the expected value is a fundamental concept that offers the average outcome of a random event when the event is repeated multiple times. Imagine you're playing a game—much like the one with silver dollars and slugs! To determine if playing the game is worth your while, you'd consider the expected value.

• The expected value is calculated by multiplying each possible outcome by its respective probability.
• You then sum these products together to get the expected result if the game were repeated extensively.
• In our game example, this calculation helped find out that each play of the game is "worth" $0.25 on average.

The concept assists in analyzing the long-term fairness or value of strategies, and in deciding on actions when faced with uncertainty. Understanding the expected value is crucial in fields ranging from gambling to economic decisions.

Fair Game

A "fair game" in probability is one in which the expected value is zero. Essentially, this means that, on average, you neither gain nor lose money over the long run.
For the game with the bag of coins, the calculation shows a fair price of $0.25 per play. This value ensures that if you play the game for a long time, you would break even.

• If the cost to play the game is equal to the expected value, the game is fair.
• In the coin game example, this means paying $0.25 for each draw to expect a zero net gain/loss over time.
• If the cost is more or less than the expected value, the game is tilted in favor of the player or the house.

Understanding the notion of a fair game helps assess not just recreational games but also investment opportunities and insurance decisions.

Probability Distribution

Probability distribution shows how likely different outcomes are within an experiment or game. It assigns a probability to each possible outcome of a random event.
The silver dollar and slug game illustrates a basic probability distribution:

• The bag contains 8 coins in total, with these probabilities:
• A silver dollar has a probability distribution of \( \frac{1}{4} \) (since there are 2 silver dollars).
• A slug has a probability distribution of \( \frac{3}{4} \) (as there are 6 slugs).

Understanding the probability distribution helps in predicting future events by knowing all possible results and their likelihoods. This forms the backbone of statistical analysis and risk assessment.

Outcome Analysis

Outcome analysis involves examining the possible results of a certain event and determining their probable impacts. By dissecting each potential outcome, we gain a clearer picture of what to expect and plan accordingly.
In our game scenario:

• Two possible outcomes exist: drawing a silver dollar or a slug.
• Outcome analysis requires understanding the value of each result— $1 for a silver dollar and $0 for a slug.
• This analysis allows us to compute the expected value, which guides in identifying a fair price to play.

Engaging in outcome analysis helps in minimizing risks and maximizing benefits across various decisions, ranging from everyday choices to complex financial strategies.