Two cards are drawn at random from a well-shuffled pack of 52 cards. Find the probability that both are red or queens.

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Keywords: Probability, cards, red cards, queens, dependent events.

Required Approach: Factual and Analytical

Points to Remember:

  • Understanding of probability calculations, specifically for dependent events.
  • Knowledge of the composition of a standard deck of cards (52 cards, 26 red, 26 black, 4 queens).
  • Application of the addition and multiplication rules of probability.

Introduction:

Probability is the branch of mathematics that deals with the likelihood of events occurring. In this problem, we are dealing with dependent events – the outcome of the first card draw affects the probability of the second draw. A standard deck of 52 playing cards consists of four suits (hearts, diamonds, clubs, spades), each with 13 cards (Ace, 2-10, Jack, Queen, King). Hearts and diamonds are red, while clubs and spades are black. There are 26 red cards and 4 queens in the deck. We need to calculate the probability of drawing two cards that are either red or queens.

Body:

1. Defining the Events:

Let A be the event that both cards are red.
Let B be the event that both cards are queens.
Let C be the event that at least one card is a red queen. (This overlaps with A and B)

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We need to find P(A ∪ B), which represents the probability that both cards are red OR both cards are queens. Since A and B are not mutually exclusive (it’s possible to draw two red queens), we use the principle of inclusion-exclusion:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

2. Calculating Individual Probabilities:

  • P(A): Probability that both cards are red.

    • Probability of drawing a red card first: 26/52 = 1/2
    • Probability of drawing a second red card, given the first was red: 25/51
    • P(A) = (26/52) * (25/51) = 25/102

  • P(B): Probability that both cards are queens.

    • Probability of drawing a queen first: 4/52 = 1/13
    • Probability of drawing a second queen, given the first was a queen: 3/51
    • P(B) = (4/52) * (3/51) = 1/221

  • P(A ∩ B): Probability that both cards are red queens.

    • Probability of drawing a red queen first: 2/52 = 1/26
    • Probability of drawing a second red queen, given the first was a red queen: 1/51
    • P(A ∩ B) = (2/52) * (1/51) = 1/1326

3. Applying the Inclusion-Exclusion Principle:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B) = (25/102) + (1/221) – (1/1326)
P(A ∪ B) ≈ 0.245 + 0.0045 + 0.00075 ≈ 0.25025

4. Alternative Approach (Considering all possibilities):

We could also calculate this by considering all possible scenarios where at least one card is red or a queen. This would involve a more complex combinatorial approach but would yield the same result.

Conclusion:

The probability of drawing two cards that are both red or both queens from a well-shuffled deck of 52 cards is approximately 0.25025 or 25.025%. This calculation involved understanding dependent probabilities, applying the inclusion-exclusion principle to account for overlapping events, and carefully considering the composition of the deck. The result highlights the importance of precise calculations in probability, particularly when dealing with dependent events. Further research could explore more complex card game probability scenarios, involving different numbers of cards drawn or specific card combinations. A deeper understanding of probability theory is crucial for various fields, including risk assessment, statistics, and game theory.

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