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Math  /  Data & Statistics

QuestionIt costs $10\$ 10 to play a dice game. For this game, two dice are rolled. If a sum greater than 10 is rolled, the player receives $30\$ 30. If a sum less than six is rolled, the player receives $25\$ 25. If a player rolls two odd numbers, then they receive $5\$ 5. A player can only receive one prize. Therefore, if a roll meets the description of more than one prize, the player only receives the higher prize value (not both). The expected value (to the nearest cent) of the game is $\$ □\square
In the long run, does the game favor the player? □\square yes □\square ✓\checkmark ) Submit Question

Studdy Solution

STEP 1

What is this asking? Figure out how much money you can expect to win or lose on average when playing this dice game. Watch out! Don't forget: You can only win the highest prize if multiple conditions are met!

STEP 2

1. Calculate probabilities
2. Determine expected winnings
3. Calculate expected value

STEP 3

First, let's **figure out the probability** of rolling a sum greater than 10.
The possible sums greater than 10 with two dice are 11 and 12.
- Sum of 11: Possible rolls are (5,6) and (6,5).
That's 2 outcomes. - Sum of 12: Only (6,6) works.
That's 1 outcome.
So, there are 2+1=32 + 1 = 3 outcomes for a sum greater than 10.

STEP 4

Next, **calculate the probability** of rolling a sum less than 6.
The possible sums less than 6 are 2, 3, 4, and 5.
- Sum of 2: Only (1,1) works.
That's 1 outcome. - Sum of 3: Possible rolls are (1,2) and (2,1).
That's 2 outcomes. - Sum of 4: Possible rolls are (1,3), (3,1), and (2,2).
That's 3 outcomes. - Sum of 5: Possible rolls are (1,4), (4,1), (2,3), and (3,2).
That's 4 outcomes.
So, there are 1+2+3+4=101 + 2 + 3 + 4 = 10 outcomes for a sum less than 6.

STEP 5

Now, let's **find the probability** of rolling two odd numbers.
The odd numbers on a die are 1, 3, and 5.
- Possible rolls: (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5).
That's 9 outcomes.

STEP 6

Finally, let's **calculate the total number of outcomes** when rolling two dice.
Each die has 6 faces, so there are 6⋅6=366 \cdot 6 = 36 possible outcomes.

STEP 7

**Calculate the probability** of winning each prize:
- Probability of sum greater than 10: 336=112\frac{3}{36} = \frac{1}{12} - Probability of sum less than 6: 1036=518\frac{10}{36} = \frac{5}{18} - Probability of two odd numbers: 936=14\frac{9}{36} = \frac{1}{4}

STEP 8

**Determine the highest prize** for overlapping conditions:
- If a sum is greater than 10, you win $30\$30. - If a sum is less than 6, you win $25\$25. - If two odd numbers are rolled, you win $5\$5.

STEP 9

**Calculate the expected winnings**:
Expected winnings=(112⋅30)+(518⋅25)+(14⋅5)\text{Expected winnings} = \left(\frac{1}{12} \cdot 30\right) + \left(\frac{5}{18} \cdot 25\right) + \left(\frac{1}{4} \cdot 5\right)

STEP 10

**Calculate the expected value** of the game by subtracting the cost to play:
Expected value=Expected winnings−10\text{Expected value} = \text{Expected winnings} - 10

STEP 11

The expected value of the game is −$0.42-\$0.42.
In the long run, the game does not favor the player.

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