Three cards are drawn at random without replacement from a standard fifty-two card deck of playing cards. What is the probability that among the three cards there is a pair or three of a kind?
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To find the probability of getting a pair or three of a kind when drawing three cards from a standard deck without replacement, we need to calculate the total number of favorable outcomes and the total number of possible outcomes.
Total number of possible outcomes:
When drawing three cards without replacement from a standard 52-card deck, the total number of possible outcomes can be calculated as follows:
There are 52 options for the first card, 51 options for the second card, and 50 options for the third card. So, the total number of possible outcomes is 52 * 51 * 50.
Total number of favorable outcomes:
To calculate the total number of favorable outcomes, let's consider two cases separately:
Case 1: Pair
To get a pair, we need to select two cards of the same rank and one card of a different rank.
There are 13 ranks in a deck, and for each rank, there are 4 cards of that rank (one from each suit).
So, the number of ways to choose two cards of the same rank is 13 * C(4, 2) (C(4, 2) represents combinations).
The remaining card can be any of the remaining 48 cards (52 - 4 - 1), so there are 48 possibilities for the third card.
Therefore, the total number of favorable outcomes for a pair is 13 * C(4, 2) * 48.
Case 2: Three of a kind
To get three of a kind, we need to select three cards of the same rank.
Similar to Case 1, there are also 13 ranks, and the number of ways to choose three cards of the same rank is 13 * C(4, 3).
There are no cards left for the third card, as we have used all four cards of the selected rank.
Therefore, the total number of favorable outcomes for three of a kind is 13 * C(4, 3).
Total number of favorable outcomes:
The total number of favorable outcomes is the sum of the number of favorable outcomes for a pair and the number of favorable outcomes for three of a kind:
Total number of favorable outcomes = 13 * C(4, 2) * 48 + 13 * C(4, 3)
Now, we can calculate the probability by dividing the total number of favorable outcomes by the total number of possible outcomes:
Probability = Total number of favorable outcomes / Total number of possible outcomes
Probability = (13 * C(4, 2) * 48 + 13 * C(4, 3)) / (52 * 51 * 50)
Calculating the values, we get:
Probability = (13 * 6 * 48 + 13 * 4) / (52 * 51 * 50)
Probability = (3744 + 52) / (132600)
Probability = 3796 / 132600
Probability ≈ 0.0286
Hence, the probability of getting a pair or three of a kind when drawing three cards without replacement from a standard deck is approximately 0.0286 or 2.86%.
To calculate the probability of getting a pair or three of a kind when three cards are drawn without replacement from a standard deck of 52 cards, we need to consider the two cases separately.
Case 1: Getting a pair
To calculate the probability of getting a pair, we need to determine the number of favorable outcomes (pairs) and the number of possible outcomes (total number of ways to choose 3 cards from the deck without replacement).
Number of favorable outcomes:
To get a pair, we can choose any rank from the 13 different ranks available and any two cards out of the four cards of that rank.
So, the number of favorable outcomes = 13 * (choose 1 out of 4) = 13 * 6 = 78.
Number of possible outcomes:
When choosing 3 cards out of a 52-card deck without replacement, we can use the formula for combinations. The number of ways to choose 3 cards from a total of 52 cards is given by "52 choose 3", which can be calculated as:
Total number of outcomes = (52 * 51 * 50) / (3 * 2 * 1) = 22,100.
Therefore, the probability of getting a pair is:
Probability = favorable outcomes / possible outcomes = 78 / 22,100 ≈ 0.0035.
Case 2: Getting three of a kind
To calculate the probability of getting three of a kind, we again need to determine the number of favorable outcomes and the number of possible outcomes.
Number of favorable outcomes:
To get three of a kind, we can choose one rank out of the 13 different ranks, and then choose any three cards out of the four cards of that rank.
So, the number of favorable outcomes = 13 * (choose 1 out of 4) = 13 * 4 = 52.
Number of possible outcomes:
Similar to the previous case, the number of ways to choose 3 cards from a 52-card deck is calculated as:
Total number of outcomes = (52 * 51 * 50) / (3 * 2 * 1) = 22,100.
Therefore, the probability of getting three of a kind is:
Probability = favorable outcomes / possible outcomes = 52 / 22,100 ≈ 0.0024.
Finally, to calculate the overall probability of getting a pair or three of a kind, we add the probabilities of the two cases:
Overall Probability = Probability of a pair + Probability of three of a kind
= 0.0035 + 0.0024
≈ 0.0059.
Therefore, the probability that among the three cards drawn there is a pair or three of a kind is approximately 0.0059 or 0.59%.