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? Select each correct answer. 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 six times, X is the number of times a 3 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.

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 with replacement six times, X is the number of times a 3 is chosen.

Step-by-step explanation:

In a binomial distribution, there are only two outcomes, or outcomes that can be reduced to 2. In the first choice, we either draw a prime number or do not draw a prime number. In the third choice, we either draw a 3 or do not draw a 3.

There must be a fixed number of trials. In the first choice, we have 5 trials; in the third option, we have 6 trials.

The trials must be independent of each other. Since the cards in the first and third options are drawn with replacement, the outcome of one trial does not influence the probability of the next trial.

The probability must be the same for every trial. This is true of the first and third options.