Question

Two dice are rolled simultaneously. if both dice show 6, then the player wins $20; otherwise the player loses the game. it costs $2.00 to play the game. what is the expected gain or loss per game?

Answer 1

When two dice are rolled in a game charging $2 to play with a $20 prize for double sixes, the expected loss per game is $1.44. This is because there is a 1/36 chance to win $18 (after the cost of playing) and a 35/36 chance to lose the $2 cost.

Step-by-step explanation:

To calculate the expected gain or loss per game when two dice are rolled, we need to find the probability of the outcome where both dice show 6 (winning) versus all other outcomes (losing). There is only one way to roll two sixes out of a total of 36 possible outcomes when rolling two six-sided dice (6 options for the first die × 6 options for the second die). Therefore, the probability of winning is 1/36 and the probability of losing is 35/36.

Now, let's calculate the expected value. If the player wins, they gain $20 (but have paid $2 to play), so the net gain is $18. If the player loses, they lose the $2 they played. The expected value (E) is calculated as follows:

E = (Probability of winning) × (Net gain when winning) + (Probability of losing) × (Loss when losing)

E = (1/36) × $18 + (35/36) × (-$2)

E = ($0.50) - ($1.94)

E = -$1.44

Thus, the expected loss per game is $1.44, which means the player is expected to lose $1.44 on average each time they play the game.