Sam bought 1 of 250 tickets selling for $2 in a game with a grand prize of $400. Was $2 a fair price to pay for a ticket to play this game?

depends what you mean by "fair price"

every lottery game or gambling patters is designed to be "unfair" for the player, how else do you think Las Vegas could exist?

expected value = (1/250(400) = 1.6

So he paid too much.

OR look at it this way:
the organizers took in 250x$2.00 = 500
but only paid out 400

To determine if $2 was a fair price to pay for a ticket, we can calculate the expected value of playing the game. The expected value represents the average amount of money one can expect to win (or lose) when playing the game.

To calculate the expected value, we need to sum up the products of each possible outcome and its probability. In this case, there are two possible outcomes: winning the grand prize of $400 or losing the $2 ticket price.

The probability of winning the grand prize can be calculated as the inverse of the total number of tickets sold (250 in this case), since there is only one grand prize and all tickets have an equal chance of winning. So, the probability of winning is 1/250.

The probability of losing the $2 ticket price can be calculated as the complement of winning, which is 1 minus the probability of winning. Therefore, the probability of losing is (1 - 1/250).

To calculate the expected value, we multiply the value of each outcome by its probability and sum them up:

Expected Value = (Probability of Winning × Value of Winning) + (Probability of Losing × Value of Losing)

Expected Value = (1/250 × $400) + ((1 - 1/250) × -$2)

Simplifying this equation will give us the expected value.

To determine if $2 was a fair price for Sam to pay for a ticket, we can calculate the expected value of playing the game.

Expected value is calculated by multiplying each possible outcome by its probability of occurring, and then summing all the products.

In this case, there are two possible outcomes: winning the grand prize of $400 with a probability of 1/250, or not winning and losing the $2 ticket price with a probability of 249/250.

Let's calculate the expected value step by step:

1. Calculate the probability of winning the grand prize:
Probability of winning = 1 / 250

2. Calculate the expected value for winning the grand prize:
Expected value for winning = Probability of winning * Grand prize value = (1 / 250) * $400 = $1.60

3. Calculate the expected value for not winning:
Expected value for not winning = Probability of not winning * Negative value of ticket price = (249 / 250) * (-$2) = -$1.98

4. Calculate the overall expected value of the game:
Expected value = Expected value for winning + Expected value for not winning = $1.60 + (-$1.98) = -$0.38

Based on the calculation, the expected value of playing the game is -$0.38. This means that, on average, for each ticket played, Sam could expect to lose $0.38.

Since the expected value is negative, it indicates that paying $2 for a ticket to play this game is not a fair price. Sam is expected to lose money on average by playing the game.