An online gambling site offers a first prize of \( \$ 50,000 \) and two second prizes of \( \$ 10,000 \) each for registered users when they place a bet. A random bet will be selected over a 24 -hour period. Three million bets are received in the contest. Find the expected winnings if you can place one registered bet of \( \$ 3 \) in the given period. Let \( x \) be the amount of the net winnings. What are the possible values of \( x \) ? \( \square \) (Use a comma to separate answers as needed. Do not use commas in the individual values. Do not include the \( \$ \) symbol in your answer.) The expected payback for a person who places one registered bet is \( \$ \square \). (Round to two decimal places as needed.)

To find the possible values of \( x \), the net winnings, we consider the outcomes of placing a single bet. 1. If you win the first prize, your winnings will be \( 50000 - 3 = 49997 \). 2. If you win one of the two second prizes, your winnings will be \( 10000 - 3 = 9997 \). 3. If you don’t win anything, your winnings will be \( 0 - 3 = -3 \). Thus, the possible values of \( x \) are \( 49997, 9997, -3 \). Now, let's calculate the expected winnings. The probabilities for the outcomes are: - First prize: \( \frac{1}{3000000} \) - Second prize (either one of two): \( \frac{2}{3000000} \) - No prize: \( \frac{2999997}{3000000} \) The expected value \( E(x) \) can be calculated as follows: \[ E(x) = (49997 \times \frac{1}{3000000}) + (9997 \times \frac{2}{3000000}) + (-3 \times \frac{2999997}{3000000}) \] Calculating each term: - First prize contribution: \( 49997 \times \frac{1}{3000000} = 0.0166657 \) - Second prizes contribution: \( 9997 \times \frac{2}{3000000} = 0.0066657 \) - No prize contribution: \( -3 \times \frac{2999997}{3000000} \approx -2.999 \) Summing these up: \[ E(x) \approx 0.0166657 + 0.0066657 - 2.999 \approx -2.9756686 \] Thus, rounding to two decimal places, the expected payback for a person who places one registered bet is approximately \( -2.98 \). To summarize: The possible values of \( x \) are \( 49997, 9997, -3 \). The expected payback is \( -2.98 \).