Question: Use the SAMPLE SPACE for a 52-card deck to find the following PROBABILITIES. Write all your answers…

Use the SAMPLE SPACE for a 52-card deck to find the following PROBABILITIES. Write all your answers as FRACTIONS in SIMPLEST FORM.

When drawing TWO CARDS WITH REPLACEMENT from a standard deck of cards, find the PROBABILITY of getting:

1. A King and a King. \( P(K \cap K) = \)

2. A King and a Queen. \( P(K \cap Q) = \)

3. A Q and a face card. \( P(Q \cap FC) = \)

4. Any ♠ and any ♥. \( P(\spadesuit \cap \heartsuit) = \)

5. A 7 and any ♠. \( P(7 \cap \spadesuit) = \)

When drawing TWO CARDS WITHOUT REPLACEMENT from a standard deck of cards, find the PROBABILITY of getting:

6. A King and a King. \( P(K \cap K) = \)

7. A King and a Queen. \( P(K \cap Q) = \)

8. A Q and a face card. \( P(Q \cap FC) = \)

9. Any ♠ and any ♥. \( P(\spadesuit \cap \heartsuit) = \)

10. A 7 and any ♠. \( P(7 \cap \spadesuit) = \)

Solution

To solve these probability problems, we need to analyze the two scenarios: with replacement and without replacement. Let’s go through them step by step. ### With Replacement 1. A King and a King \( P(K \cap K) \): The probability of drawing a King on the first draw is \(\frac{4}{52}\), since there are 4 Kings in a 52-card deck. The probability of drawing a King on the second draw is also \(\frac{4}{52}\), because the first card is replaced. \[ P(K \cap K) = \frac{4}{52} \times \frac{4}{52} = \frac{1}{169} \] 2. A King and a Queen \( P(K \cap Q) \): The probability of drawing a King is \(\frac{4}{52}\). The probability of drawing a Queen is \(\frac{4}{52}\). \[ P(K \cap Q) = \frac{4}{52} \times \frac{4}{52} = \frac{1}{169} \] 3. A Q and a face card \( P(Q \cap FC) \): The probability of drawing a Queen is \(\frac{4}{52}\). There are 12 face cards (Jack, Queen, King in each suit). \[ P(Q \cap FC) = \frac{4}{52} \times \frac{12}{52} = \frac{12}{676} = \frac{3}{169} \] 4. Any ♠ and any ♥ \( P(♠ \cap ♥) \): The probability of drawing a ♠ is \(\frac{13}{52}\). The probability of drawing a ♥ is \(\frac{13}{52}\). \[ P(♠ \cap ♥) = \frac{13}{52} \times \frac{13}{52} = \frac{1}{16} \] 5. A 7 and any ♣ \( P(7 \cap ♣) \): The probability of drawing a 7 is \(\frac{4}{52}\). The probability of drawing a ♣ is \(\frac{13}{52}\). \[ P(7 \cap ♣) = \frac{4}{52} \times \frac{13}{52} = \frac{1}{52} \] ### Without Replacement 6. A King and a King \( P(K \cap K) \): The probability of drawing a King on the first draw is \(\frac{4}{52}\). After drawing one King, there are 3 Kings left in 51 cards. \[ P(K \cap K) = \frac{4}{52} \times \frac{3}{51} = \frac{1}{221} \] 7. A King and a Queen \( P(K \cap Q) \): The probability of drawing a King first is \(\frac{4}{52}\). After drawing a King, the probability of drawing a Queen is \(\frac{4}{51}\). \[ P(K \cap Q) = \frac{4}{52} \times \frac{4}{51} = \frac{4}{663} \] 8. A Q and a face card \( P(Q \cap FC) \): The probability of drawing a Queen is \(\frac{4}{52}\). After drawing a Queen, there are 11 face cards left in 51 cards. \[ P(Q \cap FC) = \frac{4}{52} \times \frac{11}{51} = \frac{11}{663} \] 9. Any ♠ and any ♥ \( P(♠ \cap ♥) \): The probability of drawing a ♠ is \(\frac{13}{52}\). After drawing a ♠, the probability of drawing a ♥ is \(\frac{13}{51}\). \[ P(♠ \cap ♥) = \frac{13}{52} \times \frac{13}{51} = \frac{13}{204} \] 10. A 7 and any ♣ \( P(7 \cap ♣) \): The probability of drawing a 7 is \(\frac{4}{52}\). After drawing a 7, the probability of drawing a ♣ is \(\frac{13}{51}\). \[ P(7 \cap ♣) = \frac{4}{52} \times \frac{13}{51} = \frac{1}{51} \]