A game costs $4 to play. A player rolls a fair six-sided die. If the player rolls a 6, the player receives $15; otherwise, the player receives nothing. Find the expected net gain or loss per game.

Step 1: Identify the possible net outcomes.

The game costs $4 to play.

If the player rolls a 6, the player receives $15. After accounting for the $4 cost, the net gain is:
$15 - $4 = $11

If the player does not roll a 6, the player receives nothing but still paid $4. The net result is:
-$4

Step 2: Find the probability of each outcome.

The probability of rolling a 6 is:
P(6) = 1/6

The probability of not rolling a 6 is:
P(not 6) = 5/6

Step 3: Recall the expected value formula.

Expected value = sum of (outcome × probability)

Step 4: Substitute the outcomes and probabilities.

E = 11(1/6) + (-4)(5/6)

Step 5: Simplify.

E = 11/6 - 20/6

E = -9/6

E = -1.50

Step 6: Interpret the expected value.

The expected net value is -$1.50 per game. The negative sign indicates an expected loss.

This does not mean the player loses exactly $1.50 on every game. Instead, over many repetitions of the game, the average net result would approach a loss of $1.50 per game.

Therefore, the expected net loss is $1.50 per game.