You are given 9 to 1 odds against rolling a sum of 8 with the roll of two fair dice, meaning you win $9 if you succeed and you lose $1 if you fail. Find the expected value (to you) of the game.

actually it is 1.25-.86

Well, when it comes to rolling dice, I can definitely roll with it! Let's calculate the expected value.

To find the expected value, we multiply the probability of each possible outcome by the amount won or lost for that outcome. So, let's break it down.

There are 36 possible outcomes when rolling two fair dice (6 sides on each die). Out of these 36 outcomes, there are 5 ways to roll a sum of 8: (2,6), (3,5), (4,4), (5,3), and (6,2).

The probability of rolling a sum of 8 is therefore 5/36. If you win, you gain $9, and if you lose, you lose $1.

So, the expected value is calculated as follows:

(Probability of winning * Amount won) + (Probability of losing * Amount lost)

Expected value = (5/36 * $9) + (31/36 * -$1)

Simplifying this equation gives us:

Expected value = $45/36 - $31/36

Expected value = $14/36

Simplifying it even further, we find:

Expected value = $7/18

Therefore, the expected value of the game for you is $7/18. Just remember, though, this is all in the realm of probability, so it's not guaranteed to be accurate. Good luck, and may the dice be ever in your favor!

To find the expected value of the game, we need to multiply the possible outcomes by their respective probabilities and sum them up.

First, let's determine the probability of rolling a sum of 8 with two fair dice. There are five ways to obtain a sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2). Since each die has six equally likely outcomes, the total number of possible outcomes is 6 x 6 = 36.

So, the probability of rolling a sum of 8 is 5/36.

Now, let's calculate the expected value:

E(X) = (probability of winning) × (amount won) + (probability of losing) × (amount lost)

E(X) = (5/36) × $9 + (31/36) × (-$1)

Simplifying:

E(X) = $45/36 - $31/36

E(X) = $14/36

Reducing the fraction:

E(X) = $7/18

Therefore, the expected value (to you) of the game is $7/18.

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

First, let's calculate the probabilities of rolling each possible sum with two dice. One way to do this is by creating a table.

| Die 1 | Die 2 | Sum |
|-------|-------|-----|
| 1 | 1 | 2 |
| 1 | 2 | 3 |
| 1 | 3 | 4 |
| 1 | 4 | 5 |
| 1 | 5 | 6 |
| 1 | 6 | 7 |
| 2 | 1 | 3 |
| 2 | 2 | 4 |
| 2 | 3 | 5 |
| 2 | 4 | 6 |
| 2 | 5 | 7 |
| 2 | 6 | 8 |
| 3 | 1 | 4 |
| 3 | 2 | 5 |
| 3 | 3 | 6 |
| 3 | 4 | 7 |
| 3 | 5 | 8 |
| 3 | 6 | 9 |
| 4 | 1 | 5 |
| 4 | 2 | 6 |
| 4 | 3 | 7 |
| 4 | 4 | 8 |
| 4 | 5 | 9 |
| 4 | 6 | 10 |
| 5 | 1 | 6 |
| 5 | 2 | 7 |
| 5 | 3 | 8 |
| 5 | 4 | 9 |
| 5 | 5 | 10 |
| 5 | 6 | 11 |
| 6 | 1 | 7 |
| 6 | 2 | 8 |
| 6 | 3 | 9 |
| 6 | 4 | 10 |
| 6 | 5 | 11 |
| 6 | 6 | 12 |

There are 36 possible outcomes, and we can see that there are 5 ways to roll a sum of 8.

Now let's calculate the expected value. The formula for the expected value is:

Expected Value = (Probability of winning * Payout for winning) + (Probability of losing * Payout for losing)

In this case, the probability of winning is 5/36 (since there are 5 favorable outcomes out of 36 possible outcomes), and the payout for winning is $9. The probability of losing is 31/36 (since there are 31 unfavorable outcomes out of 36 possible outcomes), and the payout for losing is -$1 (losing $1).

Expected Value = (5/36 * $9) + (31/36 * -$1)
Expected Value = $45/36 - $31/36
Expected Value = $14/36
Expected Value = $0.39 (rounded to two decimal places)

Therefore, the expected value of the game to you is $0.39.

Given prob of not rolling 8 is 9/10

then prob of rolling 8 = 1/10

NOTE: with a roll of 2 fair dice, the above given probability is not correct,
prob(rolling an 8) is 5/36 and prob (not rolling a sum of 8 is 31/36

But, going with your data anyway, ....

expected value = (1/10)(9) - (9/10)(1) = 0

REAl expected value
= (5/36)(9) - (31/36)(1) = 19/9 or appr 2.11