Final answer:
The expected payback for the described game is an average loss of about $1.20 per game, indicating that it would not be profitable to play this game over the long term.
Step-by-step explanation:
The expected payback for a game in which a person bets $2 on a number between 0 and 250, where they win $200 if their number comes up, is calculated using the concept of expected value in probability. Expected value is computed by multiplying each possible outcome by its probability and then summing all these values.
In this game, the chance of winning is 1 in 251 (since there are 251 numbers from 0 to 250), and the payout for winning is $200. The chance of losing is 250 out of 251, and the loss is the cost of the game, which is $2. Thus, the expected value of playing this game is calculated as follows:
(1/251) * $200 + (250/251) * (-$2) = $0.7968 - $1.9920
= -$1.1952 approximately
Therefore, if you play this game repeatedly, over a long string of games, the expected value indicates an expected average loss of approximately $1.20 per game, indicating that this game would not be a good way to win money.