Answer :
Answer: $3
Step-by-step explanation:
To find the expected payoff of the game where you get paid $3 times the sum of the numbers on two dice, we need to calculate the average payoff over all possible outcomes of rolling two dice.
1. Calculate the sum of the numbers on two dice: The sum can range from 2 (if both dice show a 1) to 12 (if both dice show a 6). The possible sums are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.
2. Determine the probability of each possible sum: For example, the probability of rolling a sum of 2 is 1/36 (since there is only one way to roll a 2: both dice showing 1), and the probability of rolling a sum of 7 is 6/36 (since there are six ways to roll a 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)).
3. Calculate the expected payoff: Multiply each possible sum by $3 (the payoff for that sum) and then multiply that result by the probability of rolling that sum. Finally, sum up all these products to get the expected payoff.
4. For example, the expected payoff for a sum of 2 would be: $3 x 2 x 1/36 = $1/6. Do this calculation for all possible sums and then add up the results to find the overall expected payoff of the game.
5. The expected payoff will give you an idea of how much you can expect to win on average in this game over many plays.
By following these steps and considering all possible outcomes, you can determine the expected payoff of the game where you get paid $3 times the sum of the numbers on two dice.
