Problem 18 - 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 plays the game many times (in other words, the price at which his expectation is zero).

Step-by-Step Solution

Answer

The fair price to play the game is $0.25.

Step 1: Understand Probabilities

There are a total of 8 items in the bag: 2 silver dollars and 6 slugs. Hence, the probability of drawing a silver dollar is given by \[p(Silver) = \frac{2}{8} = \frac{1}{4}.\]The probability of drawing a slug is\[p(Slug) = \frac{6}{8} = \frac{3}{4}.\]

Step 2: Calculate Expected Value

To find the fair price, we need to calculate the expected value of the game. The payoff for drawing a silver dollar is \(1 (since the player keeps the dollar), and the payoff for drawing a slug is \)0 (since slugs are worthless). Thus, the expected value \(E\) of the game is calculated as follows:\[E = (1)\cdot p(Silver) + (0)\cdot p(Slug) = (1)\cdot \frac{1}{4} + (0)\cdot \frac{3}{4} = \frac{1}{4}.\]

Step 3: Determine Fair Price

The fair price to play the game is the expected value of the game because it ensures that over a long period, the average payout equals the average cost. Therefore, the fair price is \(\frac{1}{4}\) or $0.25.

Key Concepts

Probability in Games of Chance

When playing games of chance, understanding probability is crucial. Imagine you have a bag with different coins, some valuable and some not. To predict your chances of drawing a valuable coin, you calculate probabilities.

In this example, there are 2 silver dollars and 6 slugs in a bag, totaling 8 items. The probability of drawing a silver dollar is:

• Number of silver dollars = 2
• Total number of coins = 8

So, the probability is \[ p(Silver) = \frac{2}{8} = \frac{1}{4} \]
The probability of drawing a slug is:

• Number of slugs = 6
• Total number of coins = 8

Thus, the probability is \[ p(Slug) = \frac{6}{8} = \frac{3}{4} \]These probabilities help determine how likely an event is to occur, making them fundamental in evaluating games like these.

Concept of a Fair Game

A fair game is one where each player has an equal chance of winning or losing in the long run. In mathematical terms, it means that the expected profit or loss over many plays should be zero.

In the bag of coins example, determining fairness involves calculating if playing the game many times would result in breaking even.

If too much is charged to play, over time, players would lose money. Conversely, if too little is charged, the game would lose money. Therefore, a fair game balances these outcomes, ensuring neither party is at a disadvantage.

Understanding Expected Payout

Expected payout is a key concept to determine what a player can expect to gain from repeated plays in a game of chance. It's calculated by weighing each outcome by its probability. This weighted average provides the expected benefit from participating in the game.

For the coin game, if the outcome when drawing a silver dollar pays 1 dollar and a slug pays nothing, the expected payout for each play is:

• Probability of a silver dollar = \( \frac{1}{4} \)
• Probability of a slug = \( \frac{3}{4} \)
• Payoff for silver dollar = 1
• Payoff for slug = 0

Calculating the expected value:\[E = (1) \cdot \frac{1}{4} + (0) \cdot \frac{3}{4} = \frac{1}{4} \]The expected payout, or the fair price to charge for the game, is \(0.25\). This ensures that over time, a player neither gains nor loses money from playing.