To find the probability of drawing a club two times in a row without replacement from a standard deck of 52 cards, we'll follow these steps:
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Calculate the probability of drawing a club on the first draw: There are 13 clubs in a deck of 52 cards. Therefore, the probability of drawing a club on the first draw is: \[ P(\text{first club}) = \frac{13}{52} = \frac{1}{4} \]
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Calculate the probability of drawing a club on the second draw, given that the first draw was a club: If the first card drawn is a club, there are now 12 clubs left in a deck of 51 cards (since one card has been removed). Therefore, the probability of drawing a club on the second draw is: \[ P(\text{second club | first club}) = \frac{12}{51} \]
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Determine the overall probability of both events happening: To find the total probability of both events (drawing a club on the first draw and then again on the second draw), we multiply the probabilities of the two individual events: \[ P(\text{two clubs in a row}) = P(\text{first club}) \times P(\text{second club | first club}) = \frac{13}{52} \times \frac{12}{51} \]
Calculating this gives: \[ P(\text{two clubs in a row}) = \frac{1}{4} \times \frac{12}{51} = \frac{12}{204} = \frac{1}{17} \]
So, the probability of drawing a club two times in a row without replacement is \(\frac{1}{17}\).
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