Final answer:
The expected value of the game is $3.96.
Step-by-step explanation:
The expected value of a game is calculated by multiplying each possible outcome by its probability and summing the results.
In this case, there are two outcomes: losing $12 if a 4 appears at least once, and winning $7 otherwise.
Probability of getting a 4 at least once: 1 - Probability of not getting a 4 three times
= 1 - (5/6) * (5/6) * (5/6)
= 1 - (125/216)
= 91/216
Amount lost when a 4 appears at least once: -$12
Probability of not getting a 4 at all:
(5/6) * (5/6) * (5/6)
= 125/216
Amount won when a 4 does not appear at all: $7
Expected value = (Probability of losing * amount lost) + (Probability of winning * amount won)
= (91/216 * -$12) + (125/216 * $7)
= -$4.22 + $8.18
= $3.96
Therefore, the expected value of the game is $3.96.