Math Problem Statement
Two cards are drawn from a standard deck of 52 cards, without replacement. Determine the following probabilities. Show your work and round answers to three decimal places.
a. P(both cards are face cards) = ____
b. P(second card is a face card given that the first card was not a face card) = ______
Two cards are drawn from a standard deck of 52 cards, without replacement. Determine the following probabilities. Show your work and round answers to three decimal places.
a. P(both cards are face cards) = ____
b. P(second card is a face card given that the first card was not a face card) = ______
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Solution
To determine the probabilities, let's first outline the relevant facts about a standard deck of 52 cards:
- There are 12 face cards in total (3 face cards in each of the 4 suits: Jack, Queen, King).
- The remaining cards are non-face cards (52 - 12 = 40).
Part (a): P(both cards are face cards)
-
Calculate the probability of drawing a face card first:
- There are 12 face cards in a 52-card deck.
- Probability of drawing a face card first:
-
Calculate the probability of drawing a face card second (after one face card has already been drawn):
- After drawing one face card, there are now 11 face cards left and 51 total cards.
- Probability of drawing a face card second:
-
Combine these probabilities:
-
Calculate the final probability:
Part (b): P(second card is a face card given that the first card was not a face card)
-
Calculate the probability that the first card is not a face card:
- There are 40 non-face cards.
- Probability of drawing a non-face card first:
-
Calculate the probability of drawing a face card second (after a non-face card has been drawn):
- The number of face cards remains the same (12), but the total number of cards is now 51.
- Probability of drawing a face card second:
-
Combine these probabilities using the conditional probability:
-
Calculate the final probability:
Final Answers
a.
b.
If you need any further details or have questions, feel free to ask! Here are some related questions you might consider:
- What is the probability of drawing at least one face card in two draws?
- How does the probability change if we draw with replacement?
- What is the probability of drawing two cards of the same suit?
- How can we generalize these probabilities for larger or smaller decks?
- What other card combinations could you analyze with similar methods?
Tip: Remember to always consider the changing total number of cards and favorable outcomes when dealing with probabilities without replacement!
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Math Problem Analysis
Mathematical Concepts
Probability
Combinatorics
Formulas
P(A and B) = P(A) * P(B|A)
P(A|B) = P(A and B) / P(B)
Theorems
Conditional Probability
Suitable Grade Level
Grades 9-12
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