Question

Determine whether the random variable X has a binomial distribution. If it does, state the number of trials N. If it does not, explain why not.

Ten cards are randomly drawn with replacement from a standard deck of 52. Let X be the number of Hearts drawn.
The random variable [has or does not have?] a binomial distribution.

Answer 1

The random variable \( X \) does have a binomial distribution in this scenario.

Here's why:

1. **Number of Trials**: The experiment consists of drawing cards 10 times, which is fixed. This corresponds to \( N = 10 \) trials.

2. **Two Possible Outcomes**: For each draw, there are two possible outcomes regarding the event of interest (drawing a Heart or not drawing a Heart).

3. **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 \( \frac{13}{52} = \frac{1}{4} \).

4. **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 \( p = \frac{1}{4} \).