To find the expectation of this game, we need to determine the probability of each outcome and the associated winnings or losses.
There are a total of 36 possible outcomes when rolling two dice (6 possibilities for the first dice and 6 possibilities for the second dice). To find the probability of getting a sum of ten, we count the number of ways this can happen. The possible outcomes are (4, 6), (5, 5), and (6, 4), so there are three favorable outcomes. Therefore, the probability of getting a sum of ten is 3/36, or 1/12.
Similarly, there are three ways to get a sum of four: (1, 3), (3, 1), and (2, 2). So, the probability of getting a sum of four is also 1/12.
The remaining outcomes have no winnings, so we don't need to calculate the probability since the winnings are zero.
To calculate the expectation, we multiply each outcome by its probability and sum them up:
Expected value = [ (10 * 1/12) + (20 * 1/12) + (0 * 23/36) ] - $5
= [$0.8333 + $1.6667 + $0] - $5
= $2.5 - $5
= -$2.5
Therefore, the expectation of this game is -$2.5, meaning the player is expected to lose an average of $2.5 per game.
If a gambler rolls two dice and gets a sum of ten, he wins $10, and if he gets a sum of four, he wins $20. The cost to play the game is $5. What is the expectation of this game?
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