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.

Question

A fair die is rolled 30 times. Let X bet the number of times an odd number appears.

The random variable [has or does not have?] a binomial distribution.

Answer 1

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

Here’s why:

1. Fixed Number of Trials (N): The die is rolled 30 times, which means there are a fixed number of trials, \( N = 30 \).

2. Two Possible Outcomes: For each roll of the die, there are only two outcomes regarding the event we are interested in: either an odd number appears (success) or it does not (failure). The odd numbers on a fair die are 1, 3, and 5, making the probability of success \( p = \frac{3}{6} = \frac{1}{2} \).

3. Independent Trials: The rolls of the die are independent of each other. The outcome of one roll does not affect the outcome of another roll.

4. Constant Probability of Success: The probability of rolling an odd number remains the same across all trials, which is \( p = \frac{1}{2} \).

Since all the criteria for a binomial distribution are met, we conclude that \( X \), the number of times an odd number appears when rolling a die 30 times, follows a binomial distribution with \( N = 30 \).

Answer 2

So the number of trials is 30?

Answer 3

Yes, that's correct! The number of trials \( N \) is 30, as the die is rolled 30 times. Therefore, the random variable \( X \) that counts the number of times an odd number appears is modeled with a binomial distribution with \( N = 30 \).