Question

Marti decides to keep placing a $1 bet on number 15 in consecutive spins of a roulette wheel until she wins. On any spin, there’s a 1-in-38 chance that the ball will land in the 15 slot. Let Y = the number of spins it takes for Marti to win. (a) Calculate and interpret the mean of Y. (b) Calculate and interpret the standard deviation of Y

Answer 1

The mean number of spins it takes for Marti to win is 38. The standard deviation of Y is approximately 36.475.

Step-by-step explanation:

The mean of a random variable is calculated by multiplying each of the possible values by their respective probabilities, and then summing up these products. In this case, the random variable Y represents the number of spins it takes for Marti to win. For each spin, the probability of winning is 1/38, and the probability of losing is 37/38. Let's calculate the mean:

Mean of Y = (1/38) * 1 + (37/38) * (1 + Mean of Y)

Solving this equation, we find that Mean of Y = 38. So, the mean number of spins it takes for Marti to win is 38.

The standard deviation of a random variable represents the average amount that each value deviates from the mean. It can be calculated using the formula:

Standard deviation of Y = sqrt[(1/38) * (1 - Mean of Y)^2 + (37/38) * (1 + Standard deviation of Y)^2]

Solving this equation, we find that Standard deviation of Y ≈ 36.475. So, the standard deviation of Y is approximately 36.475.

Answer 2

To calculate the mean of Y, find the probability of winning in one spin and take the reciprocal. The mean of Y is 38, so on average, Marti will need to spin the wheel 38 times before she wins. To calculate the standard deviation of Y, find the variance and take the square root. The standard deviation represents the typical number of additional spins Marti needs to make before she wins, beyond the mean of 38 spins.

Step-by-step explanation:

In a roulette wheel, there is a 1-in-38 chance that the ball will land in the 15 slot. Let Y be the number of spins it takes for Marti to win. To calculate the mean of Y:

1. Find the probability of winning in one spin: 1/38
2. The expected value, or mean, of Y is equal to the reciprocal of the probability of winning in one spin: 38
3. Interpretation: On average, Marti will need to spin the wheel 38 times before she wins.

To calculate the standard deviation of Y:

1. The variance of Y is equal to the reciprocal of the probability of winning in one spin squared: (1/38)^2
2. The standard deviation of Y is the square root of the variance: √((1/38)^2)
3. Interpretation: The standard deviation tells us the average amount of deviation or spread from the mean. In this case, it represents the typical number of additional spins Marti needs to make before she wins, beyond the mean of 38 spins.