What is the probability of drawing a king and a queen consecutively from a deck of 52 cards without replacement? Select one: α. 0.45 b. 0.7 c. 0.6 d. 0.55
The Correct Answer and Explanation is :
To calculate the probability of drawing a king and a queen consecutively from a standard 52-card deck without replacement, let’s break it down step by step.
Step 1: Understand the problem
- A deck has 52 cards, with 4 kings and 4 queens.
- Since the cards are drawn without replacement, the total number of cards reduces after the first draw.
- The order matters because “king and then queen” is specified.
Step 2: Compute the probability
- Probability of drawing a king first:
[
P(\text{King first}) = \frac{\text{Number of kings}}{\text{Total cards}} = \frac{4}{52}
] - Probability of drawing a queen second:
After drawing a king, 51 cards remain, and there are still 4 queens in the deck:
[
P(\text{Queen second | King first}) = \frac{\text{Number of queens}}{\text{Remaining cards}} = \frac{4}{51}
] - Overall probability:
Multiply the probabilities of the two events since they are dependent:
[
P(\text{King and then Queen}) = P(\text{King first}) \times P(\text{Queen second | King first})
]
[
P(\text{King and then Queen}) = \frac{4}{52} \times \frac{4}{51} = \frac{16}{2652} = \frac{4}{663} \approx 0.00603
]
Step 3: Match the options
None of the provided options (( \alpha = 0.45 ), ( b = 0.7 ), ( c = 0.6 ), ( d = 0.55 )) are correct, as the actual probability (( \approx 0.006 )) is much smaller. This may indicate an issue with the options in the question.
Explanation
The probability is low because drawing a specific combination of two cards in a particular order from a deck of 52 cards is a rare event. With only 4 kings and 4 queens, the chances diminish further as the cards are drawn without replacement.