Problem 8 - Algebra and Trigonometry

Question

\(3-12\) . Find the expected value (or expectation) of the games described. A coin is tossed twice. Albert wins \(\$ 2\) for each heads and must pay \(\$ 1\) for each tails.

Step-by-Step Solution

Answer

The expected value of Albert's winnings is $1.

Step 1: Understanding the Problem

Albert wins $2 for each heads and loses $1 for each tails in two coin tosses. We need to find the expected value of his winnings.

Step 2: Identify Possible Outcomes

Since a coin is tossed twice, the possible outcomes are: HH, HT, TH, and TT. These outcomes mean: - HH: Heads on both tosses. - HT: Heads first, then tails. - TH: Tails first, then heads. - TT: Tails on both tosses.

Step 3: Assign Probabilities

Each of the outcomes has the same probability since the coin is fair. Thus: - Probability of HH = 1/4 - Probability of HT = 1/4 - Probability of TH = 1/4 - Probability of TT = 1/4.

Step 4: Calculate Albert's Winnings for Each Outcome

Calculate Albert's net winnings for each outcome: - HH: (2*2 - 0*1) = $4 - HT: (2*1 - 1*1) = $1 - TH: (2*1 - 1*1) = $1 - TT: (0*2 - 2*1) = -$2.

Step 5: Compute Expected Value

The expected value is calculated by multiplying each outcome's value with its probability and summing them up:\[ (1/4)(4) + (1/4)(1) + (1/4)(1) + (1/4)(-2) \].

Step 6: Simplify the Expression

Calculate each part separately and add them:\[ 1/4 * 4 = 1, \]\[ 1/4 * 1 = 0.25, \]\[ 1/4 * 1 = 0.25, \]\[ 1/4 * (-2) = -0.5. \]Now add them all: 1 + 0.25 + 0.25 - 0.5 = 1.

Key Concepts

Probability Theory

Probability theory is a mathematical framework used to quantify the likelihood of different outcomes. It is essential for determining how likely various scenarios are, especially when dealing with uncertain events, like tossing a coin. In the context of our exercise, each outcome of a coin toss experiment is assigned a probability.
For a coin, fair means that both heads and tails have the same chance of appearing. Hence, the probability for each possible outcome out of multiple trials is equally distributed. Here, the coin is tossed twice, and there are four possible discrete outcomes: HH, HT, TH, and TT. Each outcome is simple and distinct with a probability of being 1/4 or 25%.

• HH (Heads on both tosses): Probability = 1/4
• HT (Heads first, then Tails): Probability = 1/4
• TH (Tails first, then Heads): Probability = 1/4
• TT (Tails on both tosses): Probability = 1/4

This even probability distribution is fundamental for calculating expectations and making predictions about the game.

Discrete Outcomes

In probability, discrete outcomes refer to finite or countable results that can occur in a situation. Unlike continuous outcomes, where results can fall anywhere on a continuum, discrete outcomes are specific and separated. In our coin-tossing exercise, discrete outcomes mean that we have clear, distinct results of each trial: HH, HT, TH, and TT.
Each of these outcomes represents different scenarios with specific results for Albert's winnings or losses. Let's break them down:

• HH: Albert gets $4 because each head gives him $2.
• HT and TH: Albert ends up with $1, winning $2 from one head and losing $1 from one tail.
• TT: Albert actually loses money, ending up with $-2 because both tosses result in tails, costing him $1 each.

The clear separation of these outcomes is crucial because it helps in quickly evaluating probabilities and expected values. It is these distinctly defined outcomes that enable precise calculations and predictions.

Mathematical Expectation

Mathematical expectation, commonly referred to as expected value, is a key concept in probability that provides a measure of the 'center' of a random variable's possible outcomes. It is the average value you can expect if you repeated an experiment many times. In simple terms, it tells us what we might expect to win or lose on average.
To find the expected value, you multiply each discrete outcome by its probability and sum these products. In Albert's game:

• Expected value from HH: \[ (1/4) \times 4 = 1 \]
• Expected value from HT: \[ (1/4) \times 1 = 0.25 \]
• Expected value from TH: \[ (1/4) \times 1 = 0.25 \]
• Expected value from TT: \[ (1/4) \times (-2) = -0.5 \]

Adding these values together gives us:\[ 1 + 0.25 + 0.25 - 0.5 = 1 \]So, the expected value, or mathematical expectation, of Albert's winnings from this coin toss game is \(1. So, on average, Albert can expect to come out with \)1 after playing this game multiple times.