Problem 3 - Algebra and Trigonometry

Question

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { The game consists of drawing a card from a deck. You win }} \\ {\$ 100 \text { if you draw the ace of spades or lose } \$ 1 \text { if you draw }} \\ {\text { any other card. }}\end{array} $$

Step-by-Step Solution

Answer

The expected value of the game is approximately $0.94.

Step 1: Define the Probability of Each Outcome

To find the expectation, we need to list the possible outcomes and determine their probabilities. In a standard deck, there are 52 cards. The probability of drawing the ace of spades is \( \frac{1}{52} \) and the probability of drawing any other card is \( \frac{51}{52} \).

Step 2: Determine the Payoff for Each Outcome

Now, we identify the payoffs. Drawing the ace of spades results in a gain of $100. Drawing any other card results in a loss of $1.

Step 3: Calculate Expected Value for Each Outcome

Calculate the expected gain (or loss) for each outcome by multiplying the probability of the outcome by the payoff. For the ace of spades, this is \( \frac{1}{52} \times 100 = \frac{100}{52} \). For any other card, this is \( \frac{51}{52} \times (-1) = -\frac{51}{52} \).

Step 4: Sum the Expected Values

The expected value of the game is the sum of the expected values of each individual outcome. This is \( \frac{100}{52} + \left(-\frac{51}{52}\right) \).

Step 5: Simplify the Expected Value

Simplify that sum: \( \frac{100}{52} - \frac{51}{52} = \frac{49}{52} \). Therefore, the expected value of the game is approximately 0.94.

Key Concepts

Probability in Games

When engaging in card games, understanding probability is crucial. Probability is a way of quantifying the likelihood of events happening. In the context of drawing a card from a deck, each card has an equal chance of being picked. For example, if you would like to find the probability of drawing a specific card like the ace of spades, you divide the number of favorable outcomes by the total number of possible outcomes.

• Favorable outcome: Drawing the ace of spades, which is just 1 card.
• Total outcomes: All cards in the deck, which are 52 cards.

This means the probability of drawing the ace of spades is \( \frac{1}{52} \). The probability helps in determining what might happen if an event, such as drawing a card, is repeated numerous times.

Understanding Card Games

Card games are not just about luck; they are a mix of skill and probability. In our exercise, the game rules are straightforward: - Draw a card from a standard deck of 52. - Win $100 if you draw the ace of spades. - Lose $1 for any other card. Card games often involve decisions made based on possible outcomes and their probabilities. Players usually focus on the cards' values and how to maximize their chances of winning. Knowledge of the deck structure, including the types and numbers of cards, significantly impacts the strategy and understanding of potential outcomes.

Expected Value Calculation Explained

The concept of expected value is essential in evaluating risks and potential rewards in games of chance like card games. Expected value is essentially the average result of an event if you were to repeat the event many times. To calculate it:1. Identify possible outcomes and their probabilities.2. Multiply each outcome by its probability.3. Sum these values to get the expected value.For instance, calculate the game's expected value as follows:- Win \(100 probability: \( \frac{1}{52} \)- Win event value: \( \frac{1}{52} \times 100 = \frac{100}{52} \)- Lose \)1 probability: \( \frac{51}{52} \)- Lose event value: \( \frac{51}{52} \times -1 = -\frac{51}{52} \)Finally, sum the expected values: \( \frac{100}{52} - \frac{51}{52} \approx 0.94 \). This means over time, you might expect to gain $0.94 per game played. It's a statistical expectation, not a guaranteed outcome.

Understanding Payoff

The payoff in any game is the reward or penalty you receive as a result of a particular outcome. In our card game exercise: - Drawing the ace of spades results in a payoff of $100. - Drawing any other card results in paying $1, indicating a loss. Payoffs are crucial in deciding whether a game, or a particular strategy in a game, is worth the risk. It involves calculating the likely monetary outcome based on the probabilities of winning or losing certain amounts. Consider the size of the payoff and the probability of the outcome occurring. In our example, even though losing is more probable, the payoff of winning makes the game intriguing. Calculating expected value helps to summarize these possibilities and their payoffs, providing a numerical insight into the game's risk and benefit. This highlights how a high payoff in unlikely outcomes can balance out lower payoffs in probable outcomes to create a potentially advantageous situation.