Question

Penalty Worksheet – Due 10-19         Expected Value of a Game        Name_______________________

Instructions:You must show your work and all work must be organized and easy to follow. You will have to make an appointment with me to discuss your work. You will not receive credit if you cannot adequately explain your work to me in person.

I pick a digit (an integer from 0 to 9, inclusive). Then, for a dollar, you get to pick three digits randomly, with replacement, by spinning a spinner (see picture below). Then the three digits you selected are compared to mine. The following describes the possible outcomes:

If none of your digits matches mine, you lose your dollar.

If exactly one of your digits matches mine, you win $1 net (your initial dollar plus one dollar from me).

If exactly two of your digits matches mine, you win $10 net.

If all three of your digits match mine, you win $100 net.

a. Let X= the number of your digits that match mine. What are the possible values of X?

b. Xis a binomial random variable. (Why?) If so, what are nand p?

c. Construct the probability distribution of X. (Remember, the distribution must list the possible values of X, and their corresponding probabilities.)

d. Let W= your net winning for one play of the game. Describe the possible values of W.

e. Construct the probability distribution of W.

f. Use the distribution of Wto find and interpret the expected value (mean) of W, i.e., E(net winnings).

g. Is the game "fair?” (What does it mean to describe the game as fair?) If not, determine how much you should pay to play the game in order to make it "fair." Show your work or explain your reasoning.

h. Suppose that, rather than selecting a number from 0 to 999 randomly, you are allowed to pick your own number between 0 and 999. Determine a strategy that optimizes your expected net winnings. Find the expected winnings for your strategy. (There is more than one optimal number, but only one optimal strategy.)

Answer 1

The RV represents the number of digits matcde out of 3. The probability of matching a digit is .

a )Hence is binomial Rv with .

b) The PMF of is .

Here .

c) The probability distribution of is

d) The possible values of W are .

e) The probability distribution of is

f) The expected value of is

g) The game is not fair since

We are required to solve only 4 parts. Please post the remaining questions as another post.
We do not get any additional amount for solving more. I have solved 7 parts.