Lottery: I buy one of 250 raffle tickets for $5. The sponsors then randomly select 1 grand prize worth $250, 2 second prizes worth $100 each, and 3 third prizes worth $50 each. Below is the discrete probability distribution for this raffle.
Prize P(x)
Grand 1/250
Second 2/250
Third 3/250
None 244/250
(a) Recognizing that I spent $5 to buy a ticket, determine the expected value of this raffle to me as a player. Round your answer to the nearest penny.
$
(b) What is an accurate interpretation of this value?
opt1- It represents how much you would win every time you play the game.
opt2- It is meaningless because you can't actually win or lose this amount.
opt3- It represents the per-game average you would win/lose if you were to play this game many many times.
opt4 - It represents how much you would lose every time you play the game.
(c) Based on your answers, would this raffle be a good financial investment for you and why? There is only one correct answer and reason.
opt1- Yes, because the expected value is positive.
opt2- Yes, because the expected value is negative.
opt3- No, because the expected value is positive.
opt4- No, because the expected value is negative.
To calculate the expected value, we multiply the value of each outcome by its corresponding probability and sum them up. Let's calculate it for this raffle:
(a) Grand prize: $250 x (1/250) = $1
Second prize: $100 x (2/250) = $0.80
Third prize: $50 x (3/250) = $0.60
None: $0 x (244/250) = $0
Expected value = $1 + $0.80 + $0.60 + $0 = $2.40
So, the expected value of this raffle for you as a player is $2.40.
(b) The accurate interpretation of this value is opt3 - It represents the per-game average you would win/lose if you were to play this game many, many times. The expected value indicates the long-term average outcome if the game is played repeatedly.
(c) Based on the above answers, the correct option is opt1 - Yes, because the expected value is positive. A positive expected value means that on average, you would expect to win money if you were to play this game many times. Therefore, this raffle would be a good financial investment for you.