Question

Pick a card, any card. You draw a card from a deck. If you get a red card, you win nothing. If you get a spade, you win \(\$ 5\). For any club, you win \(\$ 10\) plus an extra \(\$ 20\) for the ace of clubs. a) Create a probability model for the amount you win. b) Find the expected amount you'll win. c) What would you be willing to pay to play this game?

Short answer

The expected win is about $4.14. Pay up to $4.14 to play.

Step-by-step solution

Understand the Deck Composition

A standard deck has 52 cards: 26 red cards, 13 spades, and 13 clubs. Among the clubs, there is 1 ace. We will use these proportions to calculate probabilities.

Define the Outcomes

Identify the possible outcomes for winnings: - Red card: Win $0 - Spade: Win $5 - Non-ace club: Win $10 - Ace of clubs: Win $30 ($10 + $20 extra)

Calculate Probabilities for Each Outcome

- Probability of a red card: \(\frac{26}{52} = \frac{1}{2}\)- Probability of a spade: \(\frac{13}{52} = \frac{1}{4}\)- Probability of a non-ace club: \(\frac{12}{52} = \frac{3}{13}\)- Probability of the ace of clubs: \(\frac{1}{52}\)

Construct the Probability Model

Based on the outcomes and probabilities, the model is:- Win \(0 with probability \(\frac{1}{2}\)- Win \)5 with probability \(\frac{1}{4}\)- Win \(10 with probability \(\frac{3}{13}\)- Win \)30 with probability \(\frac{1}{52}\)

Calculate the Expected Value

The expected value \(E\) is calculated as:\[E = (0) \cdot \frac{1}{2} + (5) \cdot \frac{1}{4} + (10) \cdot \frac{3}{13} + (30) \cdot \frac{1}{52}\]Simplifying the arithmetic, \[E = 0 + 1.25 + \frac{30}{13} + \frac{30}{52}\]Convert fractions to decimals: \[E \approx 0 + 1.25 + 2.31 + 0.5769 \approx 4.1369\]So, the expected value of winning is approximately $4.14.

Decide the Entry Fee

You should be willing to play the game for no more than the expected value, so the maximum reasonable entry fee to play the game should be approximately $4.14.

Key concepts

Expected Value

The concept of expected value is a fundamental aspect of probability theory. It helps predict the average outcome if an experiment or game is repeated many times. In our card game, it comprises various possible winnings, each with its own probability.

To calculate the expected value, you multiply each possible winning (or losing) amount by its probability and sum them all up. This calculation gives a sense of the 'average' gain or loss per play over a long period.

• For our card game, we have four different outcomes: winning $0, $5, $10, or $30.
• Each outcome has a specific probability based on the card's color or suit.
• The expected value is essentially your return per play if you play the game an infinite number of times.

The calculated expected value of around $4.14 indicates, on average, how much you would win per game in the long run.

Deck Composition

Understanding the composition of a deck is crucial for solving probability-related problems. A standard deck has 52 cards divided into four suits: hearts, diamonds, clubs, and spades, each containing 13 cards.

For this exercise, extraction of deck information enables the creation of our probability model. It helps us see the relations between the total number of cards and desired outcomes.

• The deck has 26 red cards (hearts and diamonds), contributing to a loss in our game scenario.
• There are 13 spades, which result in a specific winning of $5.
• Similarly, among the clubs, one card—the ace—offers an enhanced winning of $30 instead of $10 for a normal club.

The knowledge of how these cards are distributed allows for accurate probability distributions, which are the basis for further calculations.

Probability Calculation

Probability calculation is the process of determining the likelihood of various outcomes. In this card game, particular attention is devoted to different hands and the probabilities of each.

We assign probabilities to each event, with the total probability always adding up to 1. For each potential winning outcome, we assess its chance based on the deck's composition.

• Probability of drawing a red card: \( P(0) = \frac{26}{52} = \frac{1}{2} \)
• Probability of drawing a spade: \( P(5) = \frac{13}{52} = \frac{1}{4} \)
• Probability of drawing a non-ace club: \( P(10) = \frac{12}{52} = \frac{3}{13} \)
• Probability of drawing the ace of clubs: \( P(30) = \frac{1}{52} \)

These probabilities are essential for creating the probability model. They quantify the chance of each outcome, and when paired with the winnings, they assist in computing the expected value.

Entry Fee Decision

Deciding on an entry fee requires understanding the relationship between risk and reward. The key is comparing the expected value of playing against what you're willing to pay.

Ideally, the entry fee should not exceed the expected value, ensuring you don't pay more than the average return. This precaution emphasizes playing games that offer a fair chance of not just participating but also potentially earning a reward.

• If the entry fee is higher than the expected value, it means you're likely to lose money over time.
• An entry fee equal to or less than the anticipated return ($4.14) would be reasonable.

This prudent decision-making ensures responsible participation, as it aligns interests with statistical odds, benefiting players by avoiding games of unfavorable expected outcomes.