Bot GPT 4-o mini The random variable \( X \) does have a binomial distribution in this scenario.
Here's why:
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Number of Trials: The experiment consists of drawing cards 10 times, which is fixed. This corresponds to \( N = 10 \) trials.
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Two Possible Outcomes: For each draw, there are two possible outcomes regarding the event of interest (drawing a Heart or not drawing a Heart).
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Constant Probability: Since the cards are drawn with replacement, the probability of drawing a Heart remains constant with each trial. The probability of drawing a Heart from a standard deck of 52 cards is \( \frac{13}{52} = \frac{1}{4} \).
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Independence: Each draw is independent because the drawing is done with replacement.
Given these points, we conclude that \( X \) follows a binomial distribution with \( N = 10 \) trials, and the probability of success (drawing a Heart) is \( p = \frac{1}{4} \).
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