Final answer:
The expected value of the carnival dice game is -$1.61, meaning you would lose an average of $1.61 per game played. This calculation is based on the probabilities of winning ($2 net gain) and losing (-$3), indicating it is not a financially favorable game.
Step-by-step explanation:
The expected value for the carnival dice game can be calculated by considering the outcomes where you win money and the cost of playing the game.
Each die has 6 faces, so there are 6 x 6 = 36 possible outcomes when rolling two dice. To find the probability of each outcome, we need to count how many combinations give us sums less than 6 (which are sums of 2, 3, 4, and 5). These combinations are (1,1), (1,2), (2,1), (1,3), (3,1), (2,2), (1,4), (4,1), (2,3), and (3,2), totaling 10 outcomes. The probability of winning is 10/36.
The game pays out $5 for winning and costs $3 to play. Thus, if you win, your net gain is $5 - $3 = $2. If you lose (26 outcomes out of 36), you simply lose the cost to play, which is $3. We use the probability of each outcome multiplied by its corresponding value to find the expected value:
Expected Value (EV) = (Probability of Winning) x (Net Gain when Winning) + (Probability of Losing) x (Loss when Losing)
EV = (10/36) x $2 + (26/36) x (-$3)
EV = (10/36) x $2 + (26/36) x (-$3) = (20/36) - (78/36) = -$58/36
EV = -$1.61 (rounded to the nearest cent)
This means that for each game played, you can expect to lose an average of $1.61 in the long run. Hence, the carnival game has a negative expected value, implying it is not a favorable game to play financially.