The Florida lottery game Lotto is played by choosing six numbers from 1 to 53. If these numbers match the ones drawn by the lottery commission, you win the jackpot. The table below shows the probability of winning a $23,000,000 jackpot, along with the amounts and probabilities of other prizes. Assuming that the jackpot was not split among multiple winners, find the expectation for buying a $1 lottery ticket. Round to the nearest tenth of a cent.
To find the expectation for buying a $1 lottery ticket, we need to calculate the expected value of winning.
The probability of winning the jackpot is 1 in 22,957,480, which gives a prize of $23,000,000.
The probability of winning $5,000 is 6 in 22,957,480, which gives a prize of $30,000.
The probability of winning $100 is 125 in 22,957,480, which gives a prize of $12,500.
The probability of winning $5 is 7,150 in 22,957,480, which gives a prize of $35,750.
The probability of winning $1 is 145,362 in 22,957,480, which gives a prize of $145,362.
The probability of not winning anything is 22,806,333 in 22,957,480, which gives a prize of $0.
To calculate the expectation, we multiply the probability of each outcome by its respective prize and sum up those values.
Expectation = (1/22,957,480 * $23,000,000) + (6/22,957,480 * $30,000) + (125/22,957,480 * $12,500) + (7,150/22,957,480 * $35,750) + (145,362/22,957,480 * $145,362) + (22,806,333/22,957,480 * $0)
Expectation = $1.004
Therefore, the expectation for buying a $1 lottery ticket is $1.004.
To find the expectation for buying a $1 lottery ticket, we need to multiply the amount of each prize by its probability and sum these values. Let's start by calculating the expected value for winning the jackpot of $23,000,000.
The probability of winning the jackpot is 1 in 22,957,480 (since there are 22,957,480 possible combinations of 6 numbers out of 53). So, the expected value for winning the jackpot is:
(1/22,957,480) * $23,000,000 = $1
Next, let's calculate the expected values for the other prizes:
- Match 5 numbers: The probability of matching 5 numbers is 1 in 103,769, and the prize amount is $5,000. So, the expected value for this prize is:
(1/103,769) * $5,000 ≈ $48.16
- Match 4 numbers: The probability of matching 4 numbers is 1 in 1,386, and the prize amount is $70. So, the expected value for this prize is:
(1/1,386) * $70 ≈ $50.45
- Match 3 numbers: The probability of matching 3 numbers is 1 in 61, and the prize amount is $5. So, the expected value for this prize is:
(1/61) * $5 ≈ $0.08
Finally, we can calculate the expectation by summing the expected values for all prizes:
$1 + $48.16 + $50.45 + $0.08 = $99.69
Therefore, the expectation for buying a $1 lottery ticket is approximately $99.7.