During a game, a balanced die is rolled. A player receives $22 when an even number is rolled, and loses $12 when an odd number is rolled. How much money can he expect on average in the long run?
Is the answer 17?
For a balanced die, the probability of getting an even number is n({2,4,6})/n({1,2,3,4,5,6})=1/2.
Similarly, the probability of getting an odd number is 1/2.
Thus, the expected gain/roll is
expected gain - expected loss
= 0.5*$22 - 0.5*$12
= $11-$6
=$5 per roll on average in the long run
To determine how much money the player can expect on average in the long run, we need to calculate the expected value.
The expected value is calculated by multiplying each possible outcome by its probability and summing up the results.
In this case, when a balanced die is rolled, there are three even numbers (2, 4, 6) and three odd numbers (1, 3, 5), making a total of six possible outcomes. Each outcome has a probability of 1/6 since there is an equal chance of rolling any number on a fair die.
When an even number is rolled, the player receives $22, so the contribution to the expected value is (22 * 1/6) = $3.67.
On the other hand, when an odd number is rolled, the player loses $12, so the contribution to the expected value is (-12 * 1/6) = -$2.
Adding up both contributions, we get ($3.67 - $2) = $1.67. This means, on average, the player can expect to make $1.67 per roll.
Therefore, the correct answer is $1.67, not $17.