We spin a roulette wheel 720 times. Find the following expectations:


# of greens
# of reds
# of spins that show 7, 14, 21, or 28

To find the expectations, we need to understand some basic concepts.

The roulette wheel consists of 38 pockets: 18 red, 18 black, and 2 green. Each spin of the wheel is considered an independent event, which means the outcome of one spin does not affect the outcomes of subsequent spins.

To find the expectations, we need to calculate the average number of occurrences of each event over the 720 spins.

1. Number of greens:
The probability of getting a green pocket in one spin is 2/38, as there are 2 green pockets out of 38 total pockets. Since the spins are independent, we can expect to get a green pocket in 2/38 of the spins. Therefore, the expected number of greens is (2/38) * 720.

Expected # of greens = (2/38) * 720 = 38

2. Number of reds:
The probability of getting a red pocket in one spin is 18/38, as there are 18 red pockets out of 38 total pockets. Therefore, the expected number of reds is (18/38) * 720.

Expected # of reds = (18/38) * 720

3. Number of spins that show 7, 14, 21, or 28:
Each of the numbers 7, 14, 21, and 28 has a probability of 1/38 of appearing in one spin. Since the spins are independent, we can calculate the expected number of spins that show these numbers by multiplying the probability of each number by 720, and then summing up the values.

Expected # of spins showing 7, 14, 21, or 28 = (1/38) * 720 + (1/38) * 720 + (1/38) * 720 + (1/38) * 720

Remember to simplify the fractions when calculating the expectations.

Expected # of spins showing 7, 14, 21, or 28 = (4/38) * 720

I hope this helps!