A gambler offers to let you play the following game: he will roll two (fair) dice. If the sum of the two dice is 7, you will lose $8. If the sum of the two dice is 2, you will lose $7. For any other roll, you will win $4. What is the expected value of this gamble?

Ah, gambling... the epitome of mathematical opportunity. Let me crunch some dicey numbers for you.

We have six possible outcomes when rolling two fair dice: (1,1), (1,2), (1,3), (1,4), (1,5), and (1,6). And that's just when the first die shows a 1! You can imagine the endless possibilities.

To determine the expected value, we multiply the payoff of each outcome by its probability and sum it all up. So let's do some calculations:

The probability of rolling a sum of 7 is 6/36 (since there are 6 ways to roll a 7 and 36 total possibilities), and the payoff is -8 dollars.

The probability of rolling a sum of 2 is 1/36, and the payoff is -7 dollars.

For any other roll (i.e., not a sum of 7 or 2), which is 29/36 of the time, you'll win 4 dollars.

Now, let's put all these values into our equation:

Expected Value = (Probability of 7 × Payoff for 7) + (Probability of 2 × Payoff for 2) + (Probability of any other roll × Payoff for any other roll)

Expected Value = (6/36 × -8) + (1/36 × -7) + (29/36 × 4)

Calculating that, we get -8/6 - 7/36 + 116/36 = -4/3 - 7/36 + 29/9 = -179/54 ≈ -3.315.

So, my friend, the expected value of this gamble is approximately -3.315 dollars. Now, can we interest you in some non-gambling activities? Like juggling flaming torches perhaps? It's slightly less hazardous to your financial well-being.

To find the expected value of this gamble, we need to calculate the probability of each outcome and multiply it by the corresponding payoff.

Let's start by calculating the probability of each outcome:

1. The sum of the two dice is 7. There are six possible combinations that yield a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). So the probability is 6/36 = 1/6.

2. The sum of the two dice is 2. There is only one combination that yields a sum of 2: (1,1). So the probability is 1/36.

3. For any other roll, the sum will be a number between 3 and 12, excluding 2 and 7. There are (36 - 6 - 1) = 29 possible combinations. So the probability is 29/36.

Next, let's calculate the expected value:

- If the sum is 7, you will lose $8. So the payoff is -8.
- If the sum is 2, you will lose $7. So the payoff is -7.
- For any other roll, you will win $4. So the payoff is +4.

Now we can calculate the expected value:

Expected value = (Probability of Sum 7 * Payoff for Sum 7) + (Probability of Sum 2 * Payoff for Sum 2) + (Probability of Any other roll * Payoff for Any other roll)

Expected value = (1/6 * (-8)) + (1/36 * (-7)) + (29/36 * 4)

Calculating this expression, we get:

Expected value = (-8/6) + (-7/36) + (116/36)

Expected value = (-48 + (-7) + 116) / 36

Expected value = 61/36

Hence, the expected value of this gamble is $1.69 (rounded to two decimal places).

To find the expected value of a gamble, you need to multiply the possible outcomes by their probabilities and sum them up. Let's calculate the expected value step by step for each possible sum of the two dice:

1. The sum of two dice can be any number from 2 to 12. However, we only need to consider the sums 2, 7, and any other sum, as these are the only outcomes that affect the winnings/losses in this game.

2. The sum of 2: This can occur in only one way since the minimum value on a die is 1. The probability of rolling a sum of 2 is 1/36, as there is only one combination that results in a sum of 2: rolling a 1 on both dice.

3. The sum of 7: This can occur in six different ways: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). The probability of rolling a sum of 7 is 6/36 or 1/6, as there are six favorable outcomes out of the total 36 possible outcomes when rolling two fair six-sided dice.

4. Any other sum: For all other possible sums, there are 36 - 7 - 1 = 28 combinations. Since all combinations have an equal probability of 1/36, the probability of rolling any other sum is 28/36 or 7/9.

Now, let's calculate the expected value using the given outcomes and their probabilities:

Expected value = [Probability of Sum of 2 * Loss on Sum of 2] + [Probability of Sum of 7 * Loss on Sum of 7] + [Probability of Any Other Sum * Win on Any Other Sum]

Expected value = (1/36 * (-7)) + (1/6 * (-8)) + (7/9 * 4)
Expected value = -7/36 - 8/6 + 28/9
Expected value = -7/36 - 48/36 + 112/36
Expected value = 57/36
Expected value ≈ $1.58

Therefore, the expected value of this gamble is approximately $1.58. This means that, on average, you can expect to win $1.58 per game when playing this gamble repeatedly in the long run.

Prob(7) = 1/6

Prob(2) = 1/36
Prob(neither a 7 nor a 2) = 1 - 1/6 - 1/36 = 29/36

expected value = -(1/6)(8.00) - (1/36)(7.00) + (29/36)(4.00)
= .....