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Chapter 5: Problem 52

Playing cards Shuffle a standard deck of playing cards and deal one card. Define events \(J\) : getting a jack, and \(R\) : getting a red card. (a) Construct a two-way table that describes the sample space in terms of events \(J\) and \(R\) (b) Find \(P(J)\) and \(P(R)\). (c) Describe the event " \(J\) and \(R\) " in words. Then find \(P(J\) and \(R)\) (d) Explain why \(P(J\) or \(R) \neq P(J)+P(R) .\) Then use the general addition rule to compute \(P(J\) or \(R)\)

Short Answer

Expert verified
(a) Two-way table constructed. (b) \( P(J) = \frac{1}{13}, P(R) = \frac{1}{2} \). (c) "\(J\) and \(R\)"= red jack, \( P(J \text{ and } R) = \frac{1}{26} \). (d) \( P(J \text{ or } R) = \frac{7}{13} \), using general addition rule.

Step by step solution

01

Define the Deck

A standard deck of playing cards has 52 cards: 4 suits (hearts, diamonds, clubs, spades) with 13 cards each. Red cards are hearts and diamonds (26 cards total). Each suit includes 1 jack, so there are 4 jacks in total: hearts, diamonds, clubs, and spades.
02

Construct the Two-Way Table

We classify the cards based on whether they are jacks and whether they are red cards:\[\begin{array}{|c|c|c|}\hline & \text{Jack} \, (J) & \text{Not Jack} \, (\bar{J}) \\hline\text{Red} \, (R) & 2 & 24 \\hline\text{Not Red} \, (\bar{R}) & 2 & 24 \\hline\end{array}\]This table represents 2 red jacks (hearts and diamonds), 24 other red cards, 2 black jacks (clubs and spades), and 24 other black cards.
03

Find P(J) and P(R)

The probability of drawing a jack, \(P(J)\), is the number of jacks over the total number of cards: \[P(J) = \frac{4}{52} = \frac{1}{13}\]The probability of drawing a red card, \(P(R)\), is the number of red cards over the total number of cards: \[P(R) = \frac{26}{52} = \frac{1}{2}\]
04

Describe Event 'J and R' and Find Its Probability

"\(J\) and \(R\)" describes drawing a card that is both a jack and red, i.e., either the jack of hearts or the jack of diamonds. There are 2 such cards. Therefore, the probability is \[P(J \text{ and } R) = \frac{2}{52} = \frac{1}{26}\]
05

Use the General Addition Rule

The probability that a card is either a jack or a red card is not simply the sum of their probabilities because they overlap (there are red jacks). We use the general addition rule:\[P(J \text{ or } R) = P(J) + P(R) - P(J \text{ and } R)\\]\[P(J \text{ or } R) = \frac{1}{13} + \frac{1}{2} - \frac{1}{26} = \frac{4}{52} + \frac{26}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13}\]

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Key Concepts

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

Two-way table
A two-way table is a great way to organize data and visualize relationships between two categorical variables. In our card example, we're considering two factors: whether the card is a jack or not, and whether the card is red or not.

The table has columns for two literal categories (Jack and Not Jack) and rows for the other two categories (Red and Not Red). It helps to map out the total 52-card deck broken down by these attributes. Specifically, the table shows:
  • 2 Red Jacks (the jack of hearts and the jack of diamonds)
  • 24 other Red cards
  • 2 Black Jacks (the jack of clubs and the jack of spades)
  • 24 other Black cards
By laying these probabilities in a two-way table, you can easily see how the events intersect and overlap. This makes it much easier to count outcomes and compute related probabilities.
Sample space
The sample space in probability represents all the possible outcomes of a certain event. When drawing a card from a standard 52-card deck, the sample space is all 52 cards. Each different card in the deck represents a unique outcome.

In the context of events like drawing a jack (J) or a red card (R), the sample space encompasses every card in the deck that could fulfill these conditions.
  • For event J, the sample event space includes all four jacks.
  • For event R, it includes all 26 red cards.
Understanding the sample space is crucial because it defines the framework within which we calculate probabilities.
Addition rule
In probability theory, the addition rule is used to find the probability of the union of two events. Specifically, it helps us determine the probability of either one event or another occurring.

The rule states: \[P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B).\]
This formula adjusts for any overlap (where both events happen simultaneously). In our card scenario, we see it in action with events J (drawing a jack) and R (drawing a red card).
  • Simply adding the probabilities of J and R would overestimate since red jacks fall into both categories.
By using the addition rule and subtracting the overlap, we calculate:\[P(J \text{ or } R) = \frac{1}{13} + \frac{1}{2} - \frac{1}{26} = \frac{7}{13}.\]
Playing cards probability
Calculating probability with playing cards is a common exercise due to the simple yet varied structure of a deck. A standard deck has 52 cards split into four suits: hearts, diamonds, clubs, and spades. Knowing the composition helps identify probabilities for various events like drawing a certain rank or color.

For example, with the event of drawing a jack (J), we know there are 4 jacks in the deck which makes \(P(J) = \frac{4}{52} = \frac{1}{13}\). For event R (getting a red card), since there are 26 red cards, \(P(R) = \frac{26}{52} = \frac{1}{2}\).
  • These ratios represent parts of the whole deck, making it easy to compute without extra tools.
Playing cards provide a practical, relatable way to explore fundamental probability concepts.

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