Final answer:
The expected winnings for placing one bet of $3 in the online gambling site is -$2.92. This considers the probabilities of winning either the first prize or one of the two second prizes, along with the possibility of not winning any prize.
Step-by-step explanation:
To find the expected winnings for someone who places one registered bet of $3 on an online gambling site, we need to understand the possible outcomes and their probabilities, then calculate the expected value (E(X)) for the net winnings (X).
The possible outcomes for the net winnings X are:
- Winning the first prize of $50,000, with a net winning of $50,000 - $3 = $49,997
- Winning one of the two second prizes of $15,000, with a net winning of $15,000 - $3 = $14,997
- Not winning any prize, which results in a net loss of the bet amount, $3.
The probabilities are:
- Probability of winning the first prize (P1) = 1/1,000,000
- Probability of winning a second prize (P2) = 2/1,000,000
- Probability of not winning (P3) = 1 - (P1 + P2)
The expected value (E(X)) is calculated as:
E(X) = (49,997 × P1) + (14,997 × P2) + (-3 × P3)
Plugging in the probabilities, we have:
E(X) = (49,997 × (1/1,000,000)) + (14,997 × (2/1,000,000)) + (-3 × (1 - (3/1,000,000)))
E(X) = 0.05 + 0.03 - 3 + (3 × 0.000003)
E(X) = 0.08 - 3 + 0.000009
E(X) = -2.91999, rounded to -2.92
Therefore, the expected payback for a person who places one registered bet is -$2.92.