Problem 6 - Algebra and Trigonometry

Question

\(3-12\) . Find the expected value (or expectation) of the games described. Tim wins \(\$ 3\) if a coin toss shows heads or \(\$ 2\) if it shows tails.

Step-by-Step Solution

Answer

The expected value of the game is $2.5.

Step 1: Define the Probabilities

Since a coin has two sides, the probability of getting heads is \( \frac{1}{2} \) and the probability of getting tails is also \( \frac{1}{2} \).

Step 2: Assign Values to Outcomes

The value Tim wins if it shows heads is \( X_H = 3 \), and if it shows tails is \( X_T = 2 \).

Step 3: Use the Formula for Expected Value

The formula for expected value is given by: \( E(X) = p_H \cdot X_H + p_T \cdot X_T \), where \( p_H \) and \( p_T \) are the probabilities of heads and tails respectively.

Step 4: Calculate Expected Value

Substitute the values into the formula: \( E(X) = \frac{1}{2} \cdot 3 + \frac{1}{2} \cdot 2 \). Simplifying this gives \( E(X) = \frac{3}{2} + 1 = 2.5 \).

Key Concepts

Understanding Probability

Probability is the measure of how likely an event is to occur. It is often expressed as a fraction or a decimal between 0 and 1. The closer the probability is to 1, the more likely the event is to occur. In the context of a coin toss, since a coin has only two sides—heads and tails—each side has an equal chance of appearing. Thus, the probability of either heads or tails is

• Probability of heads = \( \frac{1}{2} \)
• Probability of tails = \( \frac{1}{2} \)

These probabilities are fundamental in calculating the expected value of a game or outcome, as they provide the basis for weighting the potential outcomes.

The Coin Toss

A coin toss is a simple, random experiment that is often used to introduce probability. Because there are only two possible results, coin tosses are a straightforward way to engage with the concept of likelihood. When a coin is tossed, it is expected to land on either heads or tails. This randomness and simplicity make the coin toss an ideal demonstration for probability concepts.
When considering games or scenarios involving coin tosses, it’s essential to understand:

• The outcome of a toss is independent of past tosses—each toss stands alone.
• The probability remains constant for each toss (\( \frac{1}{2} \) for heads, \( \frac{1}{2} \) for tails).

These characteristics are particularly important when analyzing games where outcomes of tosses might determine winnings or scores, like in Tim's game.

Outcome Values in Games

Outcome values represent the rewards or consequences associated with the results of random events, such as a coin toss. In games of chance, these values are crucial in assessing the player's expected returns. For Tim’s game, the outcome values are as follows:

• If the coin shows heads, Tim wins \(3, thus \( X_H = 3 \).
• If the coin shows tails, Tim wins \)2, thus \( X_T = 2 \).

The expected value combines these outcome values with their respective probabilities to calculate the average expected return. Using the formula:\[ E(X) = p_H \cdot X_H + p_T \cdot X_T \]we substitute the known values to find the expected value:
\[ E(X) = \frac{1}{2} \cdot 3 + \frac{1}{2} \cdot 2 = 2.5 \]
This result indicates that on average, Tim expects to win $2.50 per coin toss, providing a vital metric in assessing the fairness or profitability of the game.