You play a game in which two dice are rolled. If a sum of 7 appears, you win $10. Otherwise you lose $2.00. If you intend to play the game for a long time should you expect to make money, lose money or come out even?

first step: what is the probability of rolling a 7?

To determine whether you can expect to make money, lose money, or come out even in the long run while playing this game, we can calculate the expected value.

The expected value is the average amount of money you can expect to win or lose on each game over a large number of trials. In this case, we will consider the probabilities of rolling each possible sum of the two dice.

When two fair six-sided dice are rolled, there are 36 equally likely outcomes since there are 6 possible outcomes for each die (6 × 6 = 36). Out of these outcomes, there are 6 ways to roll a sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

Therefore, the probability of rolling a sum of 7 is 6/36 or 1/6 (since there are 6 favorable outcomes out of 36 total outcomes).

Now, let's calculate the expected value:

Expected value = (Probability of winning × Amount won) + (Probability of losing × Amount lost)

Expected value = (1/6 × $10) + (5/6 × -$2)

Expected value = ($10/6) + (-$10/6)

Expected value = $10/6 - $10/6

Expected value = $0

The calculated expected value is zero. This means that, in the long run, you can expect to come out even while playing this game. Although you might win $10 occasionally, the losses from the other outcomes will balance it out, resulting in no net profit or loss over a large number of games.