To solve this, we need to calculate the expected outcome of playing this game 839 times.

Step 1: Determine the probability of winning

When you roll two dice, the possible sums are between 2 and 12. The sum of 8 or more means sums of 8, 9, 10, 11, or 12. To find the probability of this outcome, we first determine how many ways each sum can be rolled:

• Sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2) — 5 outcomes
• Sum of 9: (3, 6), (4, 5), (5, 4), (6, 3) — 4 outcomes
• Sum of 10: (4, 6), (5, 5), (6, 4) — 3 outcomes
• Sum of 11: (5, 6), (6, 5) — 2 outcomes
• Sum of 12: (6, 6) — 1 outcome

So, the total number of favorable outcomes (sum ≥ 8) is:

$5 + 4 + 3 + 2 + 1 = 15$

There are 36 possible outcomes when two dice are rolled (since each die has 6 sides, there are $6 \times 6 = 36$ possible combinations).

Therefore, the probability of rolling a sum of 8 or more is:

$P(\text{win}) = \frac{15}{36} = \frac{5}{12} \approx 0.4167$

Step 2: Determine the probability of losing

The probability of losing (rolling a sum less than 8) is the complement of the probability of winning:

$P(\text{lose}) = 1 - P(\text{win}) = 1 - 0.4167 = 0.5833$

Step 3: Calculate expected value per game

Now, we calculate the expected value per game:

• If you win: You gain $19, so the expected gain is $19 \times 0.4167 = 7.9163$
• If you lose: You lose $17, so the expected loss is $-17 \times 0.5833 = -9.9161$

Thus, the expected value per game is:

$E = 7.9163 + (-9.9161) = -2.000$

This means you can expect to lose $2 per game on average.

Step 4: Calculate expected outcome over 839 games

Now, if you play the game 839 times, your expected total win or loss is:

$\text{Total expected loss} = -2.00 \times 839 = -1678.00$

So, you can expect to lose $1,678.00 after playing the game 839 times.

Final Answer:

You would expect to lose $1,678.00.

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Would you like any further details or have any questions about this?

Here are 5 related questions:

1. What is the expected value of a similar game, but with different win/loss amounts?
2. How do you calculate the probability of different sums when rolling two dice?
3. What are the odds of rolling a sum of 12 with two dice?
4. How does the expected value help in making decisions for repeated games?
5. How can the concept of expected value apply to other types of betting games?

Tip: When calculating expected value for any game, always break down the probabilities and potential outcomes for each possible event, and then weigh them accordingly.