Each person takes turns rolling two dice. If the sum is odd, the person playing odds gets points equal to the sum of the roll. If the sum is even, the person playing evens gets points equal to the sum of the roll. Note that the points earned is independent of who is rolling the dice.
If Jessica is challenged to a game of Sums, which statement is accurate in every aspect in guiding her to the correct choice of choosing to play odds or evens?
A. E(odds) will be more because the probability for each odd number being rolled is greater. Therefore, Jessica should play odds.
B. E(evens) will be more because the value of the even numbers on the dice are more. Therefore, Jessica should play evens.
C. E(evens) = E(odds) because the different probabilities and values end up balancing out, creating a fair game. Therefore, Jessica may choose whichever she likes.
D. E(evens) will be more because there are more even numbers that result from rolling two dice. Therefore, Jessica should play evens.
Answer (1)
Correct the probabilities for sums from rolling two dice.
Calculate the expected value for odds: E ( o dd s ) = 3.5
Calculate the expected value for evens: E ( e v e n s ) = 3.5
Since E ( o dd s ) = E ( e v e n s ) , the game is fair, and Jessica can choose either. Therefore, E ( e v e n s ) = E ( o dd s ) .
Explanation
Analyze the problem We are tasked with determining whether Jessica should choose to play odds or evens in a game where two dice are rolled, and points are awarded based on whether the sum is odd or even. To make this decision, we need to calculate the expected value of playing odds and the expected value of playing evens.
Correct the probabilities First, we need to correct the probabilities provided in the table, as they contain errors. The correct probabilities for the sums of two dice rolls are as follows:
P(2) = 1/36
P(3) = 2/36
P(4) = 3/36
P(5) = 4/36
P(6) = 5/36
P(7) = 6/36
P(8) = 5/36
P(9) = 4/36
P(10) = 3/36
P(11) = 2/36
P(12) = 1/36
Calculate the expected value of odds Next, we calculate the expected value of playing odds. This is the sum of each odd sum multiplied by its probability:
E ( o dd s ) = 3 × 36 2 + 5 × 36 4 + 7 × 36 6 + 9 × 36 4 + 11 × 36 2
E ( o dd s ) = 36 6 + 20 + 42 + 36 + 22 = 36 126 = 3.5
Calculate the expected value of evens Now, we calculate the expected value of playing evens. This is the sum of each even sum multiplied by its probability:
E ( e v e n s ) = 2 × 36 1 + 4 × 36 3 + 6 × 36 5 + 8 × 36 5 + 10 × 36 3 + 12 × 36 1
E ( e v e n s ) = 36 2 + 12 + 30 + 40 + 30 + 12 = 36 126 = 3.5
Compare expected values and conclude Comparing the expected values, we find that E(odds) = 3.5 and E(evens) = 3.5. Since the expected values are equal, the game is fair, and Jessica may choose whichever she likes.
Examples
This type of probability calculation is used in various games of chance to determine if the game is fair. Casinos and game developers use these calculations to ensure that the house doesn't have an unfair advantage, or to understand the potential payouts for players. Understanding expected value helps players make informed decisions about whether to participate in a game.
