The random variable \( X \) does indeed have a binomial distribution. Here's the reasoning:
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Fixed Number of Trials: You are drawing cards 10 times (ten cards are randomly drawn), which means the number of trials \( N \) is 10.
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Two Possible Outcomes: For each draw, the outcome can either be a Heart or not a Heart. This satisfies the condition of having two possible outcomes.
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Independent Trials: The cards are drawn with replacement, meaning the outcome of one draw does not affect the outcome of another draw. Each draw is independent of the others.
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Constant Probability: The probability of drawing a Heart remains constant across each trial. In a standard deck of 52 cards, there are 13 Hearts, so the probability \( p \) of drawing a Heart on any given trial is \( \frac{13}{52} = \frac{1}{4} \).
Since all the criteria for a binomial distribution are met, we can say:
The random variable \( X \) has a binomial distribution with the number of trials \( N = 10 \).
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