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 these problems, you can use the following steps:
1. For each case, consider the available pockets and the successful outcomes required for the given condition.
2. Find the probability for each successful outcome considering the total number of pockets.
3. If there are multiple spins as in both problems, multiply the probabilities obtained in step 2 for each of those spins.
Now let's solve each problem with these steps.
Problem 1: Probability that the ball lands in a black pocket in both spins
Step 1: There are 18 black pockets and the ball must land in one of those pockets in both spins.
Step 2: The probability for each successful outcome in one spin = number of black pockets / total number of pockets = 18/38.
Step 3: Since there are 2 spins, multiply the probabilities obtained for each spin: (18/38)*(18/38) = 324/1444 ≈ 0.224.
So the probability that the ball lands in a black pocket in both spins is about 0.224.
Problem 2: Probability that the ball lands in the same pocket in all three spins
Step 1: There are 38 pockets and the ball must land in one of those pockets in all three spins.
Step 2: The probability for each successful outcome in one spin = 1/38 (since the ball must land in a specific pocket).
Step 3: Since there are 3 spins, multiply the probabilities obtained for each spin: (1/38)*(1/38)*(1/38) = 1/54872 ≈ 0.000018.
So the probability that the ball lands in the same pocket in all three spins is about 0.000018.
To solve these probability problems, we need to calculate the probability of each event occurring and then combine them using the rules of probability.
Let's set up each problem step-by-step:
1. Probability of landing in a black pocket in both spins:
Since there are 18 black pockets out of 38 total pockets, the probability of landing in a black pocket on a single spin is 18/38. Since each spin is independent, we can multiply the probabilities:
P(black pocket on 1st spin) * P(black pocket on 2nd spin)
= (18/38) * (18/38)
Multiply the fractions to get the result.
2. Probability of landing in the same pocket in all three spins:
To solve this, we need to consider that each individual spin is independent of the others. The probability of landing in a specific pocket on a single spin is 1/38 since there are 38 pockets.
P(same pocket on all three spins) = P(same pocket on 1st spin) * P(same pocket on 2nd spin) * P(same pocket on 3rd spin)
= (1/38) * (1/38) * (1/38)
Multiply the fractions to get the result.
By following these steps, you can set up and solve probability problems involving multiple spins of a roulette wheel.