Suppose in a carnival game, there are six identical boxes, one of which contains a prize. A contestant wins the prize by selecting the box containing it. Before each game, the old prize is removed and another prize is placed at random in one of the six boxes. Is it appropriate to use the binomial probability distribution to find the probability that a contestant who plays the game five times wins exactly twice? Check each of the requirements of a binomial experiment.

Yes, the five trials are independent, repeated under the same conditions, have only two outcomes, and have the same probability of success.

No, the five trials are independent, repeated under the same conditions; have only two outcomes but they do not have the same probability of success.

Yes, the five trials are dependent, repeated under the same conditions, have only three outcomes, and have the same probability of success.

No, the five trials are dependent, repeated under the same conditions; have only two outcomes, but they do not have the same probability of success.

No, the five trials are independent, have only two outcomes, and have the same probability of success, but are not repeated under the same conditions.

You are playing a game with 3 prizes hidden behind 5 doors. One prize is worth $100, another is worth $20 and another $10. You have to pay $20 if you choose a door with no prize.What is your expected winning?

The appropriate answer is:

No, the five trials are independent, repeated under the same conditions; have only two outcomes but they do not have the same probability of success.

Explanation:
To use the binomial probability distribution, a series of trials must meet specific requirements:

1. Independent trials: Each trial's outcome should not affect the outcome of any other trial. In this case, the removal of the old prize and placement of a new prize are considered independent actions, as they do not influence each other.

2. Fixed number of trials: There are five trials in this scenario, where a contestant plays the game five times.

3. Two outcomes: In this carnival game, there are two possible outcomes - either the contestant wins the prize, or they do not.

4. Constant probability of success: To use the binomial probability distribution, each trial must have the same probability of success. However, in this game, the probability of selecting the box with the prize changes with each trial. The likelihood of success decreases as one box is removed, and the prize is placed randomly in one of the remaining boxes.

Therefore, the trials do not have the same probability of success, making it inappropriate to use the binomial probability distribution in this case.