Problem 11 - Algebra and Trigonometry

Question

\(3-12\) . Find the expected value (or expectation) of the games described. A bag contains two silver dollars and eight slugs. You pay 50 cents to reach into the bag and take a coin, which you get to keep.

Step-by-Step Solution

Answer

The expected value of the game is \(-0.30\), meaning you lose 30 cents on average per play.

Step 1: Understand the Total Number of Coins

First, recognize that there are 10 coins in total in the bag. These coins comprise 2 silver dollars and 8 slugs.

Step 2: Determine the Probability of Each Outcome

Calculate the probability of drawing a silver dollar, which is \( \frac{2}{10} = 0.2 \), and the probability of drawing a slug, which is \( \frac{8}{10} = 0.8 \).

Step 3: Calculate the Payoff for Each Outcome

Consider the winnings for each type of coin. If you draw a silver dollar, you gain a dollar, but you paid 50 cents to play, resulting in a net gain of 50 cents. Thus, the net payoff if you draw a silver dollar is \(+0.50\). If you draw a slug, you gain nothing but lose the 50 cents you paid to play, resulting in a net payoff of \(-0.50\).

Step 4: Calculate the Expected Value

Use the formula for expected value: \( E(X) = (p_1 \times v_1) + (p_2 \times v_2) \). Here \( p_1 = 0.2 \), \( v_1 = 0.50 \), \( p_2 = 0.8 \), and \( v_2 = -0.50 \). Substitute these into the formula: \[E(X) = (0.2 \times 0.50) + (0.8 \times -0.50)\]Simplify: \[E(X) = 0.10 + (-0.40) = -0.30\] So, the expected value of the game is \(-0.30\).

Key Concepts

Probability in Games of Chance

Probability is the foundation of understanding likelihoods in games of chance and more. It gives us the chance of different outcomes occurring.When dealing with probability, we evaluate how likely an event is to happen. In our given game scenario, where you draw one coin from a bag, each coin can potentially be a silver dollar or a slug.

• The probability of drawing a silver dollar is calculated as the number of silver dollars over the total coins, which is: \( \frac{2}{10} = 0.2 \).
• The probability of drawing a slug coin is \( \frac{8}{10} = 0.8 \).

Understanding these probabilities helps in predicting and evaluating the expected outcome of drawing from the bag. Keep in mind, probabilities range from 0 (impossible event) to 1 (certain event). The sum of probabilities of all possible outcomes is always 1.

Net Payoff and its Importance

When you're playing a game that involves payments or winnings, net payoff is a crucial concept. It represents the actual gain or loss you end up with after considering any costs. In our game:

• If you draw a silver dollar, you receive $1. After subtracting the 50 cents you paid to play, your net gain is 50 cents, or \(+0.50\).
• If you draw a slug, you win nothing but have still paid the entry cost of 50 cents, resulting in a net loss of 50 cents, or \(-0.50\).

The net payoff reflects the real-life impact of each possible outcome, helping you comprehend what you'll actually have after playing the game.

Understanding Random Variables

A random variable is a numerical description of the outcomes of a random phenomenon. In simpler terms, it assigns a number to the outcome of a random process, letting us calculate quantities like expected value. In this context, our random variable \(X\) represents the net payoff when you draw a coin:

• With probability \(0.2\), \(X\) is \(+0.50\) (from drawing a silver dollar).
• With probability \(0.8\), \(X\) is \(-0.50\) (from drawing a slug).

Understanding the random variable is key to computing the expected value, which tells you what to anticipate "on average" when making numerous draws from the bag in the long run, considering all pros and cons systematically. This approach transforms uncertainty into more manageable and predictable statistical insights.