Question

Keisha decides to roll a number cube until she gets a 6. Let X = the number of rolls until she rolls a 6.

What is the mean of X?
What is the standard deviation of X?

Answer 1

Explanation:

The mean of X, which represents the average number of rolls until Keisha gets a 6, can be calculated using the formula for the mean of a geometric distribution:

Mean = 1/p

Where p is the probability of rolling a 6 on a single roll of the number cube. Since there is a 1/6 probability of rolling a 6 on each roll, the mean can be calculated as:

Mean = 1/(1/6) = 6

So, the mean of X is 6.

The standard deviation of X, which measures the spread or variability of the number of rolls until Keisha gets a 6, can be calculated using the formula for the standard deviation of a geometric distribution:

Standard deviation = sqrt((1-p)/p^2)

Where p is again the probability of rolling a 6 on a single roll. Using the same probability of 1/6, the standard deviation can be calculated as:

Standard deviation = sqrt((1-(1/6))/(1/6)^2) = sqrt(5/36) ≈ 0.6455

So, the standard deviation of X is approximately 0.6455.