10. In a game of “chuck a luck” three dice are rolled. If your number comes up on all three dice you win $3. If your number comes up on two dice you win $2. If your number comes up only once you win $1. If your number doesn’t come up at all you lose $1.
a) What is the expected value of this game?
To determine the expected value of the game, we need to calculate the probability of each possible outcome and then multiply it by the corresponding amount of money won or lost.
In this game, there are 6 possible outcomes for each dice roll (numbers 1 to 6). Let's denote your chosen number as "X".
First, let's calculate the probability of getting your number on all three dice (win $3):
The probability of getting your number on a single dice roll is 1/6. Since you're rolling three dice, the probability of getting your number on all three is (1/6) * (1/6) * (1/6) = 1/216.
Next, let's calculate the probability of getting your number on two dice (win $2):
The probability of getting your number on exactly two dice out of three can be calculated in three different ways: (1/6) * (1/6) * (5/6), (1/6) * (5/6) * (1/6), or (5/6) * (1/6) * (1/6). Each of these combinations gives a probability of 5/216.
Lastly, let's calculate the probability of getting your number on only one dice (win $1):
The probability of getting your number on exactly one dice out of three can be calculated in three different ways: (1/6) * (5/6) * (5/6), (5/6) * (1/6) * (5/6), or (5/6) * (5/6) * (1/6). Each of these combinations gives a probability of 25/216.
The probability of your number not coming up at all is the complement of all the previous probabilities, which is 1 - (1/216 + 15/216 + 25/216) = 175/216.
Now that we have calculated the probabilities for each outcome, we can calculate the expected value:
Expected value = (probability of outcome 1 * amount won/lost for outcome 1) + (probability of outcome 2 * amount won/lost for outcome 2) + (probability of outcome 3 * amount won/lost for outcome 3) + (probability of outcome 4 * amount won/lost for outcome 4)
Expected value = (1/216 * $3) + (15/216 * $2) + (25/216 * $1) + (175/216 * -$1)
Expected value = $3/216 + $30/216 + $25/216 - $175/216
Expected value = ($3 + $30 + $25 - $175)/216
Expected value = -$117/216
Therefore, the expected value of this game is -$117/216.