Here are the answers and explanations for each part of the question:
a. The random variable, X, in this situation is:
The amount you win on any given roll.
b. The expected value of this game is: $1.69.
Here's how to calculate it:
Probability of winning $5: 10/36 (5 outcomes for a sum greater than 8: 5+6, 4+6, 6+4, 6+5, 6+6)
Probability of winning $1: 6/36 (2 outcomes for a sum of 7: 6+1, 1+6 and 4 outcomes for a sum of 8: 5+3, 3+5, 4+4, 2+6, 6+2)
Probability of winning $0: 20/36 (all other outcomes where the sum is less than 7)
Expected value = (5 * 10/36) + (1 * 6/36) + (0 * 20/36) = $1.69
c. The standard deviation of this game is: $2.65.
Calculating the standard deviation involves finding the variance (the average of the squared deviations from the expected value) and then taking the square root of the variance.
d. If your friend charged you $1 to play this game, I would not play.
The expected value of $1.69 is less than the cost of $1 to play.
In the long run, you would expect to lose money by playing this game repeatedly.
Complete the question:
Your friend wants you to play the following game: you roll two standard dice and compute the sum of the numbers rolled. If the sum is greater than 8, you win $5. If the sum is 7 or 8, you win $1. If the sum is less than 7, you win nothing.
a. What is the random variable, X, in this situation?- The numbers on die #1; The numbers on die #2; The sum of the dice; The amount you win on any given roll; How many times a person plays the game
b. Find the expected value of this game. - $0; $0.56; $1.69; $2.00; none
c. Find the standard deviation of this game. - $1.45; $2.09; $2.65; $4.38; none
d. If your friend charged you $1 to play this game, would you play? - No, $1 is more than I’m expected to win on any given play; No, $1 is less than I’m expected to win on any given play; Yes, $1 is more than I’m expected to win on any given play; Yes, $1 is less than I’m expected to win on any given play; Yes, $1 is equal to what I’m expected to win on any given play