To determine whether the random variable \(X\) (the number of Aces that appear when six cards are drawn with replacement from a standard deck) has a binomial distribution, we need to check the following conditions for a binomial distribution:
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Fixed number of trials: The experiment involves a fixed number of independent trials. In this case, six cards are drawn, so \(N = 6\).
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Two possible outcomes: Each trial has only two possible outcomes—success (drawing an Ace) or failure (drawing a non-Ace).
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Constant probability of success: The probability of success must remain the same for each trial. Since we are drawing with replacement from a standard deck of 52 cards, the probability of drawing an Ace (4 Aces in the deck) remains constant for each draw. The probability of drawing an Ace is \(\frac{4}{52} = \frac{1}{13}\).
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Independent trials: The outcome of each draw is independent of the others, as the draws are made with replacement.
Since all of these conditions are satisfied, we conclude that the random variable \(X\) does have a binomial distribution.
Thus, the random variable is a binomial distribution with the number of trials \(N = 6\).
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