Roulette wheels in Nevada have 38 pockets. They are all numbered 0, 00, and 1 through 36. Of all 38 pockets, there are 18 red, 18 are black, and 2 are green. Each time the wheel is spun, a ball lands in one of the pockets, and each pocket is equally likely.

If you spin the wheel twice, what is the probability lands in a black pocket in both spins?

If you spin the wheel three times, what is the probability that the ball lands in the same pocket in all three spins?

How do I set up these problems?

To set up probability problems like these, you need to consider the total number of possible outcomes and the specific outcomes you are interested in.

For the first question, we want to find the probability of landing in a black pocket in both spins. We know that there are 18 black pockets out of a total of 38 pockets. To find the probability, we divide the number of favorable outcomes (landing in a black pocket) by the total number of possible outcomes (total number of pockets). Since each spin of the wheel is independent, we multiply the individual probabilities together.

The setup for the first question would be:

P(landing in a black pocket in both spins) = P(landing in a black pocket on the first spin) * P(landing in a black pocket on the second spin)

P(landing in a black pocket in both spins) = 18/38 * 18/38

For the second question, we want to find the probability of landing in the same pocket for all three spins. Since there are no restrictions on which pocket it can be, any of the 38 pockets are favorable outcomes. Again, we use the same principle of multiplying individual probabilities since each spin is independent.

The setup for the second question would be:

P(landing in the same pocket in all three spins) = P(landing in the same pocket on the first spin) * P(landing in the same pocket on the second spin) * P(landing in the same pocket on the third spin)

P(landing in the same pocket in all three spins) = 1/38 * 1/38 * 1/38

To solve these problems, you can use the concept of probability. Probability is the measure of the likelihood of an event occurring. To set up these problems, you can start by listing all the possible outcomes and determining the desired outcomes within them.

For the first problem, you want to find the probability that the ball lands in a black pocket in both spins. The total number of possible outcomes for each spin is 38, since there are 38 pockets on the wheel. Out of these 38 pockets, 18 are black.

To find the probability of landing in a black pocket on the first spin, you divide the number of desired outcomes (18) by the total number of possible outcomes (38):

Probability of landing in a black pocket on the first spin = 18/38

Now, since each spin is independent, to find the probability of landing in a black pocket on both spins, you multiply the probabilities of each event together:

Probability of landing in a black pocket in both spins = (18/38) * (18/38)

For the second problem, you want to find the probability that the ball lands in the same pocket in all three spins. Again, the total number of possible outcomes for each spin is 38. Since you want the ball to land in the same pocket for all three spins, you need all three spins to land in one specific pocket. There are 38 possibilities for the first spin, but only 1 specific pocket is desired. Therefore, the probability for the first spin is 1/38.

Similarly, for the second and third spins, the probability is also 1/38 for each spin.

To find the probability of the ball landing in the same pocket in all three spins, you multiply the probabilities of each event together:

Probability of landing in the same pocket in all three spins = (1/38) * (1/38) * (1/38)