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An ordinary deck of 52 playing cards is randomly divided into 4 piles of 13 cards each. The probability that each pile has exactly 1 ace is
- a)0.105
- b)0.215
- c)0.516
- d)0.001
Correct answer is option 'A'. Can you explain this answer?
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To find the probability that each pile has exactly 1 ace, we can break down the problem into several steps.
Step 1: Finding the total number of ways to distribute the cards
There are 52 cards in a deck, and we need to distribute them into 4 piles of 13 cards each. The total number of ways to do this can be calculated using combinatorics. We can write it as:
Total number of ways = C(52, 13) * C(39, 13) * C(26, 13) * C(13, 13)
Where C(n, r) represents the combination of n items taken r at a time.
Step 2: Finding the number of favorable outcomes
In order for each pile to have exactly 1 ace, we need to distribute the 4 aces among the 4 piles. The first pile can have any of the 4 aces, the second pile can have any of the remaining 3 aces, and so on. So, the number of favorable outcomes can be calculated as:
Number of favorable outcomes = 4 * 3 * 2 * 1
Step 3: Calculating the probability
The probability can be calculated by dividing the number of favorable outcomes by the total number of ways. So, the probability is:
Probability = Number of favorable outcomes / Total number of ways
= (4 * 3 * 2 * 1) / (C(52, 13) * C(39, 13) * C(26, 13) * C(13, 13))
Simplifying the expression, we get:
Probability = 24 / (C(52, 13) * C(39, 13) * C(26, 13) * C(13, 13))
Using a calculator or software, we can evaluate this expression to get the final probability. The correct answer is option 'A', which is 0.105.
In conclusion, the probability that each pile has exactly 1 ace is 0.105.
Step 1: Finding the total number of ways to distribute the cards
There are 52 cards in a deck, and we need to distribute them into 4 piles of 13 cards each. The total number of ways to do this can be calculated using combinatorics. We can write it as:
Total number of ways = C(52, 13) * C(39, 13) * C(26, 13) * C(13, 13)
Where C(n, r) represents the combination of n items taken r at a time.
Step 2: Finding the number of favorable outcomes
In order for each pile to have exactly 1 ace, we need to distribute the 4 aces among the 4 piles. The first pile can have any of the 4 aces, the second pile can have any of the remaining 3 aces, and so on. So, the number of favorable outcomes can be calculated as:
Number of favorable outcomes = 4 * 3 * 2 * 1
Step 3: Calculating the probability
The probability can be calculated by dividing the number of favorable outcomes by the total number of ways. So, the probability is:
Probability = Number of favorable outcomes / Total number of ways
= (4 * 3 * 2 * 1) / (C(52, 13) * C(39, 13) * C(26, 13) * C(13, 13))
Simplifying the expression, we get:
Probability = 24 / (C(52, 13) * C(39, 13) * C(26, 13) * C(13, 13))
Using a calculator or software, we can evaluate this expression to get the final probability. The correct answer is option 'A', which is 0.105.
In conclusion, the probability that each pile has exactly 1 ace is 0.105.
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An ordinarydeck of 52 playing cards is randomly divided into 4 piles of 13 cards each. The probability that each pile has exactly 1 ace isa)0.105b)0.215c)0.516d)0.001Correct answer is option 'A'. Can you explain this answer?
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