Questions: Suppose it costs 8 to roll a pair of dice. You get paid 1 times the sum of the numbers that appear on the dice. Is it a fair game? because the expected value of the game (without considering the cost) is (Round to the nearest cent as needed.)

Suppose it costs 8 to roll a pair of dice. You get paid 1 times the sum of the numbers that appear on the dice. Is it a fair game?
because the expected value of the game (without considering the cost) is (Round to the nearest cent as needed.)

Transcript text: Suppose it costs $\$ 8$ to roll a pair of dice. You get paid $\$ 1$ times the sum of the numbers that appear on the dice. Is it a fair game? $\square$ because the expected value of the game (without considering the cost) is $\$$ $\square$ (Round to the nearest cent as needed.)

Solution

Solution Steps

Step 1: Determine the possible outcomes of rolling two dice

When rolling two dice, each die has 6 faces, so there are $ 6 \times 6 = 36 $ possible outcomes. The sum of the numbers on the dice can range from 2 to 12.

Step 2: Calculate the probability of each possible sum

The probabilities for each sum are as follows:

Sum of 2: $ \frac{1}{36} $
Sum of 3: $ \frac{2}{36} $
Sum of 4: $ \frac{3}{36} $
Sum of 5: $ \frac{4}{36} $
Sum of 6: $ \frac{5}{36} $
Sum of 7: $ \frac{6}{36} $
Sum of 8: $ \frac{5}{36} $
Sum of 9: $ \frac{4}{36} $
Sum of 10: $ \frac{3}{36} $
Sum of 11: $ \frac{2}{36} $
Sum of 12: $ \frac{1}{36} $

Step 3: Calculate the expected value of the game (without considering the cost)

The expected value $ E $ is calculated as: \[ E = \sum (\text{Sum} \times \text{Probability of that sum}) \] Substitute the values: \[ E = 2 \times \frac{1}{36} + 3 \times \frac{2}{36} + 4 \times \frac{3}{36} + 5 \times \frac{4}{36} + 6 \times \frac{5}{36} + 7 \times \frac{6}{36} + 8 \times \frac{5}{36} + 9 \times \frac{4}{36} + 10 \times \frac{3}{36} + 11 \times \frac{2}{36} + 12 \times \frac{1}{36} \] Simplify the calculation: \[ E = \frac{2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12}{36} = \frac{252}{36} = 7 \] The expected value of the game (without considering the cost) is $ \$7 $.