Questions: If you are dealt 3 cards from a shuffled deck of 52 cards, find the probability that all 3 cards are picture cards. The probability is (Round to six decimal places as needed.)

Solution Steps

To find the probability that all 3 cards dealt from a shuffled deck of 52 cards are picture cards (Jack, Queen, or King), we need to follow these steps:

1. Determine the total number of picture cards in a deck. There are 3 picture cards (Jack, Queen, King) in each of the 4 suits, so there are 12 picture cards in total.
2. Calculate the number of ways to choose 3 picture cards from these 12.
3. Calculate the number of ways to choose any 3 cards from the 52 cards in the deck.
4. The probability is the ratio of the number of ways to choose 3 picture cards to the number of ways to choose any 3 cards.
5. Round the result to six decimal places.

In a standard deck of 52 cards, there are 12 picture cards (3 per suit: Jack, Queen, King). Thus, we have: \[ \text{Total Picture Cards} = 12 \]

The number of ways to choose 3 picture cards from the 12 available is given by the combination formula: \[ \binom{12}{3} = 220 \]

The total number of ways to choose any 3 cards from the 52 cards in the deck is: \[ \binom{52}{3} = 22100 \]

The probability \( P \) that all 3 cards drawn are picture cards is the ratio of the number of ways to choose 3 picture cards to the number of ways to choose any 3 cards: \[ P = \frac{\binom{12}{3}}{\binom{52}{3}} = \frac{220}{22100} \approx 0.009955 \]

Final Answer