There are 10 cards. Each card has one number between 1 and 10, so that every number from 1 to 10 appears once.

In which distributions does the variable X have a binomial distribution?

When a card is chosen at random with replacement six times, X is the number of times a 3 is chosen.

When a card is chosen at random with replacement five times, X is the number of times a prime number is chosen.

When a card is chosen at random without replacement three times, X is the number of times an even number is chosen.

When a card is chosen at random with replacement multiple times, X is the number of times a card is chosen until a 5 is chosen.

To determine whether the variable X has a binomial distribution in each of the given scenarios, we need to check if the conditions for a binomial distribution are satisfied.

A binomial distribution has the following properties:

1. Fixed number of trials: There is a predetermined number of trials, denoted by n.

2. Independent trials: The outcome of each trial is independent of the others.

3. Constant probability of success: The probability of success (denoted by p) remains the same for each trial.

4. Binary outcome: Each trial has only two possible outcomes - success or failure.

Let's apply these conditions to each scenario:

Scenario 1: Choosing a card with replacement six times, X is the number of times a 3 is chosen.

- Fixed number of trials: We have a fixed number of trials (n = 6).
- Independent trials: Each card chosen is independent of the others.
- Constant probability of success: The probability of choosing a 3 is 1/10, and it remains constant for each trial.
- Binary outcome: Each trial can be considered as a success (choosing a 3) or a failure (not choosing a 3).

Therefore, in scenario 1, X has a binomial distribution.

Scenario 2: Choosing a card with replacement five times, X is the number of times a prime number is chosen.

- Fixed number of trials: We have a fixed number of trials (n = 5).
- Independent trials: Each card chosen is independent of the others.
- Constant probability of success: The probability of choosing a prime number depends on the distribution of prime numbers among the 10 cards. As long as the probability remains constant for each trial, this condition is satisfied.
- Binary outcome: Each trial can be considered as a success (choosing a prime number) or a failure (not choosing a prime number).

Therefore, in scenario 2, X has a binomial distribution.

Scenario 3: Choosing a card without replacement three times, X is the number of times an even number is chosen.

- Fixed number of trials: We have a fixed number of trials (n = 3).
- Independent trials: Since cards are chosen without replacement, the outcome of each trial depends on the previous selections, violating the condition of independent trials. Therefore, this scenario does not result in a binomial distribution.

Therefore, in scenario 3, X does not have a binomial distribution.

Scenario 4: Choosing a card with replacement multiple times, X is the number of times a card is chosen until a 5 is chosen.

- This scenario does not fit the standard binomial distribution setup since the number of trials is not fixed in advance. Instead, it is determined by the number of times needed to select a card with a specific outcome (5 in this case).

Therefore, in scenario 4, X does not have a binomial distribution.

In summary, X has a binomial distribution in scenarios 1 and 2, but not in scenarios 3 and 4.

The variable X has a binomial distribution in the following distributions:

1. When a card is chosen at random with replacement six times, and X is the number of times a 3 is chosen. This is because each card chosen is independent of the previous choices, and there are only two possible outcomes for each choice (getting a 3 or not getting a 3).

2. When a card is chosen at random with replacement five times, and X is the number of times a prime number is chosen. Similar to the previous case, each card chosen is independent, and there are only two possible outcomes for each choice (getting a prime number or not getting a prime number).

On the other hand, X does not have a binomial distribution in the following distributions:

3. When a card is chosen at random without replacement three times, and X is the number of times an even number is chosen. In this case, the probabilities of getting an even number change for each card chosen since the pool of cards decreases. Therefore, the outcomes are not independent, which is a requirement for a binomial distribution.

4. When a card is chosen at random with replacement multiple times, and X is the number of times a card is chosen until a 5 is chosen. Since the number of trials is not fixed and depends on when the first 5 appears, this does not fit the definition of a binomial distribution either.

In a binomial distribution, the probability of the event happening (and not happening) must remain constant in all trials. This occurs when the removed card is replaced, and not when the removed card remains unreplaced.