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 calculate probabilities in these scenarios, you need to understand the concept of probability and how it relates to the total number of possible outcomes.
For the first question, "What is the probability that the ball lands in a black pocket in both spins?", you need to find the probability of landing in a black pocket on the first spin, and then multiply it by the probability of landing in a black pocket on the second spin.
1. Determine the probability of landing in a black pocket on the first spin:
- Since there are 18 black pockets out of 38 total pockets, the probability of landing in a black pocket on the first spin is 18/38.
2. Determine the probability of landing in a black pocket on the second spin:
- Since each spin of the wheel is independent, the probability of landing in a black pocket on the second spin is also 18/38.
3. Multiply the probabilities together:
- To find the probability of both events occurring, you multiply the individual probabilities together: (18/38) * (18/38) = 0.219.
Therefore, the probability of the ball landing in a black pocket in both spins is approximately 0.219, or 21.9%.
For the second question, "What is the probability that the ball lands in the same pocket in all three spins?", you need to consider the probability of landing in any specific pocket on the first spin, and then multiply it by the probability of landing in the same pocket on the second and third spins.
1. Determine the probability of landing in the same pocket on the first spin:
- Since there are 38 pockets in total, the probability of landing in a specific pocket is 1/38.
2. Determine the probability of landing in the same pocket on the second and third spins:
- Again, because each spin is independent, the probability of landing in the same pocket on the second and third spins is also 1/38.
3. Multiply the probabilities together:
- To find the probability of all three events occurring, you multiply the individual probabilities together: (1/38) * (1/38) * (1/38) = 0.00000402.
Therefore, the probability of the ball landing in the same pocket in all three spins is approximately 0.00000402, or 0.000402%.