Question

You roll a die, winning nothing if the number of spots is odd, $1 for a 2 or a 4, and$10 for a 6.

a) Find the expected value and standard deviation of your prospective winnings.
b) You play twice. Find the mean and standard deviation of your total winnings.
c) You play 40 times. What’s the probability that you win at least $100?

Answer 1

The expected value is 1.67 and the standard deviation is 3.73 for a single roll. For two rolls, the mean is 3.34 and the standard deviation is 5.28. The probability of winning at least $100 in 40 rolls can be calculated using the binomial distribution.

Step-by-step explanation:

To find the expected value, we multiply each possible outcome by its respective probability and sum them up.

In this case, the probabilities are:

P(odd) = 3/6 = 1/2

P(2 or 4) = 2/6 = 1/3

P(6) = 1/6

The respective winnings for each outcome are:

W(odd) = 0

W(2 or 4) = 1

W(6) = 10

So the expected value is:

E(X) = (1/2)(0) + (1/3)(1) + (1/6)(10) = 1.67

The formula for the standard deviation is:

sqrt(variance) = sqrt(E(X^2)-E(X)^2)

To find the variance, we need to find E(X^2). The respective winnings squared are:

W(odd)^2 = 0

W(2 or 4)^2 = 1

W(6)^2 = 100

So E(X^2) = (1/2)(0) + (1/3)(1) + (1/6)(100) = 16.67

Now we can calculate the variance:

variance = E(X^2)-E(X)^2 = 16.67 - (1.67)^2 = 13.89

Finally, the standard deviation is:

sqrt(variance) = sqrt(13.89) = 3.73