Boris has a bag with 8 balls numbered 1 through 8 . He is playing a game of chance. This game is this: Boris chooses one ball from the bag at random. He wins $\$ 1$ if the number 1 is selected, $\$ 2$ if the number 2 is selected, $\$ 5$ if the number 3 is selected, $\$ 6$ if the number 4 is selected, $\$ 8$ if the number 5 is selected, and $\$ 10$ if the number 6 is selected. He loses $\$ 13$ if 7 or 8 is selected. (If necessary, consult a list of formulas.) (a) Find the expected value of playing the game. Ddollars (b) What can Boris expect in the long run, after playing the game many times? (He replaces the ball in the bag each time.) Boris can expect to gain money. He can expect to win $\square$ dollars per selection. Boris can expect to lose money. He can expect to lose $\square$ dollars per selection. Boris can expect to break even (neither gain nor lose money).

Step 1 ：The game Boris is playing involves selecting a ball from a bag that contains 8 balls numbered 1 through 8. The amount of money Boris wins or loses depends on the number on the ball he selects. He wins \$1 if he selects ball number 1, \$2 if he selects ball number 2, \$5 if he selects ball number 3, \$6 if he selects ball number 4, \$8 if he selects ball number 5, and \$10 if he selects ball number 6. However, he loses \$13 if he selects either ball number 7 or 8.

Step 2 ：The expected value of a random variable is calculated by multiplying each possible outcome by its probability and then summing these products. In this case, the possible outcomes are the amounts of money Boris can win or lose, and the probabilities are the chances of each amount being won or lost.

Step 3 ：Since Boris replaces the ball in the bag each time, the probability of each outcome is the same, which is \(\frac{1}{8} = 0.125\).

Step 4 ：By multiplying each outcome by its probability and then summing these products, we can calculate the expected value of playing the game. The outcomes and their probabilities are as follows: \(1 \times 0.125 = 0.125\), \(2 \times 0.125 = 0.25\), \(5 \times 0.125 = 0.625\), \(6 \times 0.125 = 0.75\), \(8 \times 0.125 = 1\), \(10 \times 0.125 = 1.25\), \(-13 \times 0.125 = -1.625\), \(-13 \times 0.125 = -1.625\).

Step 5 ：Adding these products together gives us the expected value of playing the game: \(0.125 + 0.25 + 0.625 + 0.75 + 1 + 1.25 - 1.625 - 1.625 = 0.75\).

Step 6 ：\(\boxed{0.75}\) is the expected value of playing the game. This means that, on average, Boris can expect to win \$0.75 per game in the long run. Therefore, Boris can expect to gain money, and he can expect to win \$0.75 per selection.