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Chapter 3: Problem 23

In a gambling game a person draws a single card from an ordinary 52 -card playing deck. A person is paid \(\$ 15\) for drawing a jack or a queen and \(\$ 5\) for drawing a king or an ace. A person who draws any other card pays \(\$ 4 .\) If a person plays this game, what is the expected gain?

Short Answer

Expert verified
The expected gain is approximately \(\$0.31\).

Step by step solution

01

Determine Probabilities

Calculate the probability of each outcome. There are 4 Jacks, 4 Queens, 4 Kings, and 4 Aces in a 52-card deck. The probability of drawing a Jack or a Queen is \( \frac{8}{52} \), the probability of drawing a King or an Ace is also \( \frac{8}{52} \), and the probability of drawing any other card is \( \frac{36}{52} \).
02

Calculate Expected Gains for Each Outcome

Determine the expected monetary gain for each possible card: For Jacks or Queens, the expected gain is \(15\), for Kings or Aces, it is \(5\), and for any other card, the expected loss is \(-4\).
03

Multiply Probabilities by Gains

Calculate each component of the expected value: - For drawing a Jack or Queen: \( \frac{8}{52} \times 15\)- For drawing a King or Ace: \( \frac{8}{52} \times 5\)- For drawing any other card: \( \frac{36}{52} \times (-4)\)
04

Sum the Expected Values

Add the calculated expected gains from each of the outcomes to find the overall expected gain: \[ \left( \frac{8}{52} \times 15 \right) + \left( \frac{8}{52} \times 5 \right) + \left( \frac{36}{52} \times (-4) \right) \]
05

Calculate and Simplify

Perform the arithmetic to find the total expected gain:- \( \frac{8}{52} \times 15 = 2.31 \)- \( \frac{8}{52} \times 5 = 0.77\)- \( \frac{36}{52} \times (-4) = -2.77\)Add these results: \[ 2.31 + 0.77 - 2.77 = 0.31 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value
Expected value is a fundamental concept in probability theory. It represents the average outcome of a random event if you were to repeat it many times. In the context of this gambling game, expected value helps us predict the average monetary gain or loss that a player might experience. To calculate the expected value, we multiply each possible outcome by its probability and sum the results. This gives us a single number that indicates the 'average' result. Knowing the expected value can help us decide whether participating in the game is likely to be profitable over time.
Gambling Game
A gambling game is any game that involves risk and chance, where participants can win or lose money based on the outcome. In this card game, a player draws a single card from a standard 52-card deck, and the result determines their win or loss based on predefined rules. In our specific example, players can win money if they draw a Jack, Queen, King, or Ace and must pay a penalty for drawing any other card. Gambling games often use similar mechanics, combining elements of luck and probability to determine outcomes.
Card Probability
Probability is a measure of how likely an event is to occur. In the card game example, we calculate the probability of drawing certain cards from the deck. For the given game:
  • The probability for drawing a Jack or a Queen is \( \frac{8}{52} \). There are 4 Jacks and 4 Queens in the deck.
  • The probability for drawing a King or Ace is \( \frac{8}{52} \). Similarly, there are 4 Kings and 4 Aces.
  • The probability for drawing any other card is \( \frac{36}{52} \). This is because there are 36 cards that are neither a Jack, Queen, King, nor Ace.
Understanding these probabilities is key to evaluating the game and predicting outcomes.
Monetary Gain Calculation
Monetary gain calculation involves determining the financial outcome based on certain events. In our card game, it means calculating how much a player could win or lose. To figure out the expected monetary gain, we need to:
  • Identify the gain or loss for each card category (e.g., $15 for Jacks/Queens, $5 for Kings/Aces, and a $-4 loss for other cards).
  • Multiply these gains by their respective probabilities.
  • Add these products together to find the overall expected gain.
By following such steps, we found that the calculated total expected gain for this game is $0.31. This indicates that on average, a player might expect to gain $0.31 per game, suggesting a slight advantage to playing if the calculations are accurate and consistent.

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Most popular questions from this chapter

A corporation is sampling without replacement for \(n=3\) firms to determine the one from which to purchase certain supplies. The sample is to be selected from a pool of six firms, of which four are local and two are not local. Let \(Y\) denote the number of nonlocal firms among the three selected. a. \(P(Y=1)\) b. \(P(Y \geq 1)\) c. \(P(Y \leq 1)\)

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