Question

You roll two fair dice. If they land with a sum of 7, you get a $10. If they land with a sum of 6 or 8, you get $5. Everything else earns $0. Which of the following would be the highest value you could set the price and the player would still win?
$8
$5
$2
$10​

Answer 1

Using the expected value, it is found that the highest value you could set the price is: $5.

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• A game is fair, that is, the player would still win, if the expected value is 0.
• The expected value is the sum of each outcome multiplied by it's probability.
• A probability is the number of desired outcomes divided by the number of total outcomes.

• For two fair dice, there are 36 total outcomes, as $6^2 = 36$.
• The charge is x.

• 6 outcomes result in a sum of 7, which are (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1), thus, $\frac{6}{36}$ probability of getting 10.
• 10 outcomes result in a sum of 6 or 8, which are (1,5), (2,4), (2,6), (3,3), (3,5), (4,2), (4,4), (5,1), (5,3) and (6,2), thus, $\frac{8}{36}$ probability of getting x.
• 36 - 14 = 22, thus $\frac{22}{36}$ probability of losing x.

Since we want the expected value to be 0:

$\frac{6}{36}(10) + \frac{8}{36}(5) - \frac{22}{36}x = 0$

$\frac{60 + 50 - 22x}{36} = 0$

$22x = 110$

$x = \frac{110}{22}$

$x = 5$

The highest value you could set the price is: $5.

A similar problem is given at brainly.com/question/24905256