In a game of dice, the probability of rolling a 12 is 1/36. The probability of rolling a 9, 10, or 11 is 9/36. The probability of rolling any other number is 26/36. If the player rolls a 12, the player wins $5. If the player rolls a 9, 10, or 11, the player wins $1. Otherwise, the player loses $1. What is the expected value of this game? If the game is played 100 times what are the expected winnings (or losses) of the player? I am at a loss where to start.

Question

In a game of dice, the probability of rolling a 12 is 1/36. The probability of rolling a 9, 10, or 11 is 9/36. The probability of rolling any other number is 26/36. If the player rolls a 12, the player wins $5. If the player rolls a 9, 10, or 11, the player wins $1. Otherwise, the player loses $1. What is the expected value of this game? If the game is played 100 times what are the expected winnings (or losses) of the player? I am at a loss where to start.

Answer 1

E(rolls 12) = 5(1/36)

E(rolls 9,10,11) = 1(9/36)
E(rolls any other ) = 1(26/36)

add them up for your expected win

What do they charge to play the game?

multiply your expected win by 100

Answer 2

To find the expected value of a game, you need to multiply each possible outcome by their respective probabilities, and then sum up the results.

In this case, we have three possible outcomes with different probabilities and winnings/losses:
- Rolling a 12: Probability = 1/36, Winnings = $5
- Rolling a 9, 10, or 11: Probability = 9/36, Winnings = $1
- Rolling any other number: Probability = 26/36, Winnings = -$1 (a loss of $1)

Let's calculate the expected value for each outcome:

For rolling a 12:
Expected Winnings = (1/36) * $5 = $5/36

For rolling a 9, 10, or 11:
Expected Winnings = (9/36) * $1 = $9/36

For rolling any other number:
Expected Winnings = (26/36) * (-$1) = -26/36

Now, adding up the expected winnings for each possible outcome:
Expected Value = ($5/36) + ($9/36) + (-26/36)

Simplifying:
Expected Value = ($5 + $9 - $26)/36 = -$12/36 = -$1/3

Thus, the expected value of this game is -$1/3 or -$0.33.

Now, if you want to find the expected winnings or losses over 100 games, you simply need to multiply the expected value per game by the number of games played:

Expected Winnings/Losses = (-$1/3) * 100 = -$100/3 = -$33.33

Therefore, if the game is played 100 times, the expected winnings for the player is -$33.33, which means they are expected to lose an average of $33.33.