A friend offers you the following game: You will roll 2 fair dice. If the sum of the two numbers obtained is 2,3,4,9,10,11,or 12 the friend will pay you $20. However, if the sum of the two numbers is 5, 6,7, or 8, you pay your friend $20. This friend points out to you that there are 7 winning numbers and 4 losing numbers. What is the expected value of this game? Is this a fair game to you?
Supposed a certain game is fair and costs $7 if you lose and has a net payoff of $9 if you win. The only possible outcomes of the game are winning and losing.
To calculate the expected value of a game, you need to multiply each outcome by its respective probability and sum the results. In this case, there are 7 winning outcomes and 4 losing outcomes.
To start, let's calculate the probability of each outcome:
- The sum of 2 can only occur if both dice show a 1. There is only one way this can happen, so the probability is 1/36.
- The sum of 3 can occur in two ways: (1,2) or (2,1). Each of these combinations is equally likely, so the probability is 2/36 = 1/18.
- Similarly, the sums of 4, 9, 10, 11, and 12 each have different combinations that can yield them. You can calculate their individual probabilities by counting the number of possible combinations and dividing by 36.
- Sum of 4: (1,3), (2,2), or (3,1) → 3/36 = 1/12
- Sum of 9: (3,6), (4,5), (5,4), or (6,3) → 4/36 = 1/9
- Sum of 10: (4,6), (5,5), or (6,4) → 3/36 = 1/12
- Sum of 11: (5,6) or (6,5) → 2/36 = 1/18
- Sum of 12: (6,6) → 1/36
- For the losing outcomes, we have 4 sums to consider, and their probabilities can be calculated in the same way:
- Sum of 5: (1,4), (2,3), (3,2), or (4,1) → 4/36 = 1/9
- Sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5/36
- Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), or (6,1) → 6/36 = 1/6
- Sum of 8: (2,6), (3,5), (4,4), (5,3), or (6,2) → 5/36
With the probabilities calculated, we can now determine the expected value:
Expected value = (Probability of winning * Amount won) - (Probability of losing * Amount lost)
In this case, you win $20 for every winning outcome and lose $20 for every losing outcome. Substituting the values into the equation:
Expected value = [(1/36 * $20) + (1/18 * $20) + (1/12 * $20) + (1/9 * $20) + (1/12 * $20) + (1/18 * $20) + (1/36 * $20)] - [(1/9 * $20) + (5/36 * $20) + (1/6 * $20) + (5/36 * $20)]
Expected value = [$0.56 + $1.11 + $1.67 + $2.22 + $1.67 + $1.11 + $0.56] - [$2.22 + $2.78 + $3.33 + $2.78]
Expected value = $8.90 - $11.11
Expected value = -$2.21
Since the expected value is negative (-$2.21), this means that, on average, you will lose $2.21 per game. Therefore, this is not a fair game for you, and it would not be advisable to play it.