A standard deck of cards consisting of 52 cards, 13 in each of 4 different suits, is shuffled, and 4 cards are drawn without replacement. What is the probability that all four cards are of a different suit?
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To find the probability that all four cards drawn are of a different suit, we need to determine the total number of possible outcomes and the number of outcomes where all four suits are different.
1. Total number of possible outcomes:
- The first card can be any of the 52 cards.
- The second card can be any of the remaining 51 cards.
- The third card can be any of the remaining 50 cards.
- The fourth card can be any of the remaining 49 cards.
Therefore, the total number of possible outcomes is 52 * 51 * 50 * 49 = 649,740.
2. Number of outcomes with all four suits different:
- The first card can be any of the 52 cards.
- The second card must be of one of the remaining 39 cards of a different suit.
- The third card must be of one of the remaining 26 cards of another different suit.
- The fourth card must be of one of the remaining 13 cards of the final different suit.
Therefore, the number of outcomes with all four suits different is 52 * 39 * 26 * 13 = 872,304.
3. Probability calculation:
The probability is given by the number of favorable outcomes (all four suits different) divided by the total number of possible outcomes.
Probability = Number of outcomes with all four suits different / Total number of possible outcomes
= 872,304 / 649,740
≈ 1.34 (rounded to two decimal places)
Hence, the probability that all four cards drawn are of a different suit is approximately 1.34.