Two cards are drawn without replacement from a well-shuffled deck of 52 cards. Determine the probability distribution of the number of face cards (i.e. Jack, Queen, King and Ace).

To determine the probability distribution of the number of face cards drawn from a standard deck of 52 cards, we first identify the total number of face cards. In a deck, there are 12 face cards: 3 face cards (Jack, Queen, King) in each of the 4 suits. The Ace is not considered a face card in this context.

Possible Outcomes

When drawing two cards without replacement, the number of face cards drawn can be 0, 1, or 2. Let's calculate the probabilities for each scenario:

1. Probability of Drawing 0 Face Cards

To find this probability, we need to calculate the chance of drawing two non-face cards:

• Number of non-face cards = 52 - 12 = 40
• Probability of first card being non-face = 40/52
• Probability of second card being non-face = 39/51

The combined probability is:

P(0 face cards) = (40/52) * (39/51) = 0.608

2. Probability of Drawing 1 Face Card

This scenario can occur in two ways: drawing a face card first or second. We calculate both:

• Face card first, non-face second: (12/52) * (40/51)
• Non-face first, face card second: (40/52) * (12/51)

Adding these probabilities gives:

P(1 face card) = (12/52) * (40/51) + (40/52) * (12/51) = 0.384

3. Probability of Drawing 2 Face Cards

For this case, both cards drawn must be face cards:

• Probability of first card being a face card = 12/52
• Probability of second card being a face card = 11/51

The probability is:

P(2 face cards) = (12/52) * (11/51) = 0.052

Summary of the Probability Distribution

The final probabilities for the number of face cards drawn are:

• 0 face cards: 0.608
• 1 face card: 0.384
• 2 face cards: 0.052

This distribution shows the likelihood of drawing face cards from a standard deck when two cards are drawn without replacement.