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 [blank] a binomial distribution

Answer 1

The random variable \( X \) does indeed have a binomial distribution. Here's the reasoning:

1. **Fixed Number of Trials**: You are drawing cards 10 times (ten cards are randomly drawn), which means the number of trials \( N \) is 10.

2. **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.

3. **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.

4. **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 \( N = 10 \).