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?
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.
To calculate the expectation of the game, we need to find the probability of each outcome and multiply it by the corresponding payout. Here's how we can calculate it step-by-step:
Step 1: Find the probability of rolling a sum of ten:
To roll a sum of ten, we can have the following combinations: (4, 6), (5, 5), and (6, 4). There are a total of 36 possible outcomes when rolling two dice, so the probability of rolling a sum of ten is 3/36 = 1/12.
Step 2: Calculate the payout for rolling a sum of ten:
If the sum of ten is rolled, the player wins $10.
Step 3: Calculate the expected value for rolling a sum of ten:
Expected value for rolling a sum of ten = Probability of rolling a sum of ten * Payout for rolling a sum of ten.
Expected value for rolling a sum of ten = (1/12) * $10 = $10/12 = $0.83 (rounded to the nearest cent).
Step 4: Find the probability of rolling a sum of four:
To roll a sum of four, we can have the following combinations: (1, 3), (2, 2), and (3, 1). So, the probability of rolling a sum of four is also 3/36 = 1/12.
Step 5: Calculate the payout for rolling a sum of four:
If the sum of four is rolled, the player wins $20.
Step 6: Calculate the expected value for rolling a sum of four:
Expected value for rolling a sum of four = Probability of rolling a sum of four * Payout for rolling a sum of four.
Expected value for rolling a sum of four = (1/12) * $20 = $20/12 = $1.67 (rounded to the nearest cent).
Step 7: Calculate the total expected value:
The player has to pay $5 to play the game, so we subtract this cost from the total expected value to find the net expectation.
Total Expected Value = (Expected value for rolling a sum of ten) + (Expected value for rolling a sum of four) - Cost to play the game.
Total Expected Value = $0.83 + $1.67 - $5 = $0.83 + $1.67 - $5 = -$2.5 (rounded to the nearest cent).
Therefore, the expectation of this game is -$2.50. This means that, on average, the player can expect to lose about $2.50 per game.
To find the expectation of this game, we need to calculate the expected value, which is the sum of each possible outcome multiplied by its corresponding probability.
First, let's determine the probabilities of getting a sum of ten or four.
To find the probability of rolling a sum of ten, we need to calculate the number of ways we can roll two dice to get ten. The possible combinations are (4, 6), (6, 4), (5, 5), (3, 7), and (7, 3), which totals to 5 possible combinations out of 36 total outcomes for rolling two dice.
So, the probability of getting a sum of ten is 5/36.
Similarly, to find the probability of rolling a sum of four, we need to calculate the number of ways we can roll two dice to get four. The possible combinations are (1, 3), (3, 1), (2, 2), which totals to 3 possible combinations out of 36 total outcomes for rolling two dice.
So, the probability of getting a sum of four is 3/36 (which simplifies to 1/12).
Now, let's calculate the expected value:
The gambler wins $10 when the sum is ten, and the probability of that happening is 5/36.
So, the contribution to the expected value from a sum of ten is: (10 * 5/36).
The gambler wins $20 when the sum is four, and the probability of that happening is 1/12.
So, the contribution to the expected value from a sum of four is: (20 * 1/12).
The expected value is calculated by summing up the contributions from each possible outcome:
Expected Value = (10 * 5/36) + (20 * 1/12) - 5.
Now, let's perform the calculations:
Expected Value = (10 * 5/36) + (20 * 1/12) - 5
= (50/36) + (20/12) - 5
= (25/18) + (20/12) - 5
= (25/18) + (5/3) - (90/18)
= (25/18) + (30/18) - (90/18)
= (25 + 30 - 90)/18
= -35/18.
Therefore, the expectation of this game is -$35/18 or -$1.94.
Note: A negative expectation implies that, in the long run, the player is expected to lose money on average.