To find the probability that both cards are black when Dharen randomly selects two cards from a standard deck of 52 cards without replacement, we can follow these steps:
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Determine the total number of cards: There are 52 cards in total, consisting of 26 black cards and 26 red cards.
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Calculating the probability for the first card: The probability of drawing a black card first is the number of black cards divided by the total number of cards: \[ P(\text{first card is black}) = \frac{26}{52} = \frac{1}{2} \]
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Calculating the probability for the second card: Once the first black card is drawn, there are now 25 black cards left and a total of 51 cards remaining: \[ P(\text{second card is black | first card is black}) = \frac{25}{51} \]
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Calculating the overall probability: To find the probability that both cards drawn are black, multiply the probabilities: \[ P(\text{both cards are black}) = P(\text{first card is black}) \times P(\text{second card is black | first card is black}) = \frac{26}{52} \times \frac{25}{51} = \frac{1}{2} \times \frac{25}{51} = \frac{25}{102} \]
Thus, the probability that both cards drawn are black is:
\[ \frac{25}{102} \]
Therefore, the correct response is:
\(\text{Start Fraction 25 over 102 End Fraction}\)