Question

you are given a free play at the following lottery game: three balls are drawn at random from among 12 balls numbered 1 through 12. players win the following prize amounts for guessing which balls are drawn: $224 for correctly guessing all three numbers. $22 for correctly guessing exactly two of the three numbers. $13 for correcting guessing exactly one of the three numbers. additionally, if the player guesses all three numbers in the order that they are drawn, the player wins another $1,053 (on top of the $224 won for guessing all three number). what is the expected (average) amount won playing this game?

Answer 1

The expected amount you can win playing this lottery game is approximately $0.14.

To calculate the expected amount you can win in this lottery game, we need to consider the probabilities of each winning scenario and their respective payouts:

Guessing all three numbers correctly in order:

Probability: This is the probability of choosing the 3 winning balls out of 12, in the exact order they are drawn. It can be calculated as:

(12 * 11 * 10) / (12 * 11 * 10) = 1/120

Payout: $224 + $1053 = $1277

Guessing all three numbers correctly (not necessarily in order):

Probability: This is the probability of choosing the 3 winning balls out of 12, regardless of the order. It can be calculated as:

(12 * 11 * 10) / (12 * 11 * 10) * 3! = 3/220

Payout: $224

Guessing exactly two numbers correctly:

Probability: This is the combination of choosing 2 correct balls out of 12 and 1 wrong ball. It can be calculated as:

(12 * 11) / (12 * 11) * 10 * 3 = 90/220

Payout: $22

Guessing exactly one number correctly:

Probability: This is the combination of choosing 1 correct ball out of 12 and 2 wrong balls. It can be calculated as:

(12 * 11) / (12 * 11) * 10 * 3 * 2 = 180/220

Payout: $13

Now, to find the expected value (average amount won), we multiply each probability by its respective payout and sum them up:

Expected Value = (1/120 * $1277) + (3/220 * $224) + (90/220 * $22) + (180/220 * $13) ≈ $0.14

Therefore, the expected amount you can win playing this lottery game is approximately $0.14. While there are larger prizes available, the low probability of winning them makes the overall expected value quite low.