Question

at this years state fair, there was a dice rolling game. if you rolled two dice and got a sum of 2 or 12, you won $17. if you rolled a 7, you won $6. any other roll was a loss. it cost $3 to play one game with one roll of the dice. what is the expectation of the game? round your answer to the nearest cent, if necessary. the expectation of this game is approximately dollar(s).

Explanation:

Step1: Calculate total number of outcomes

When rolling two dice, each die has 6 possible outcomes. So the total number of outcomes when rolling two dice is $6\times6 = 36$.

Step2: Find probability and net - gain for sum of 2 or 12

The combinations for a sum of 2 are $(1,1)$ and for a sum of 12 are $(6,6)$. So there are 2 favorable outcomes. The probability $P(2\text{ or }12)=\frac{2}{36}$. The net - gain when getting a sum of 2 or 12 is $17 - 3=14$ dollars.

Step3: Find probability and net - gain for sum of 7

The combinations for a sum of 7 are $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$, so there are 6 favorable outcomes. The probability $P(7)=\frac{6}{36}$. The net - gain when getting a sum of 7 is $6 - 3 = 3$ dollars.

Step4: Find probability and net - gain for other sums

The number of outcomes for other sums is $36-(2 + 6)=28$. The probability $P(\text{other})=\frac{28}{36}$. The net - gain when getting other sums is $0 - 3=-3$ dollars.

Step5: Calculate the expected value

The formula for the expected value $E$ of a discrete random variable is $E=\sum_{i}x_ip_i$. Here, $E = 14\times\frac{2}{36}+3\times\frac{6}{36}+(-3)\times\frac{28}{36}$.
\[

$$\begin{align*} E&=\frac{28 + 18-84}{36}\\ &=\frac{46 - 84}{36}\\ &=\frac{-38}{36}\approx - 1.06 \end{align*}$$

\]

Answer:

$-1.06$