Many lottery games for very large prizes include guessing which numbers will appear from a random sample of the numbered balls. In Minnesota’s Gopher 5 lottery, there are five balls numbered 1-47. If a player matches at least two balls, she wins a prize (see table below). If the cost of playing is $ t least how large must the jackpot be before the expected return of playing Gopher 5 becomes favorable to the player? Assume there is only one winner, and also ignore the fact that winnings are also taxed. Please provide just a numerical value for your answer, do not include a dollar sign. MatchWin Odds 4 of 5 $500 $15 $1 5 of 5 Jackpot 1 in 1,533,939 1 in 7,304 1 in 178 1 in 13 3 of 53 2 of 5
The Correct Answer and Explanation is :
To determine the minimum jackpot size at which the expected return from playing Minnesota’s Gopher 5 lottery becomes favorable to the player, we need to calculate the expected value (EV) of a single ticket purchase and find the jackpot amount that makes this EV equal to or greater than the ticket price of $1.
Understanding the Prize Structure and Odds:
In Gopher 5, players select five numbers from 1 to 47. The prize tiers, corresponding odds, and prize amounts are as follows:
- Match 5 of 5 (Jackpot): Odds of 1 in 1,533,939.
- Match 4 of 5: Odds of 1 in 7,304, with a prize of $500.
- Match 3 of 5: Odds of 1 in 178, with a prize of $15.
- Match 2 of 5: Odds of 1 in 13, with a prize of $1.
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Calculating the Expected Value (EV):
The EV is calculated by multiplying the probability of each outcome by its respective prize and summing these products:
- Jackpot Prize:
- Probability: 1 / 1,533,939
- Prize: $J (the jackpot amount)
- Contribution to EV: (1 / 1,533,939) * J
- Match 4 of 5:
- Probability: 1 / 7,304
- Prize: $500
- Contribution to EV: (1 / 7,304) * 500
- Match 3 of 5:
- Probability: 1 / 178
- Prize: $15
- Contribution to EV: (1 / 178) * 15
- Match 2 of 5:
- Probability: 1 / 13
- Prize: $1
- Contribution to EV: (1 / 13) * 1
Summing these contributions:
EV = (1 / 1,533,939) * J + (1 / 7,304) * 500 + (1 / 178) * 15 + (1 / 13) * 1
For the expected return to be favorable, EV must be at least equal to the ticket price of $1:
1 ≤ (1 / 1,533,939) * J + (1 / 7,304) * 500 + (1 / 178) * 15 + (1 / 13) * 1
Solving for the Jackpot Amount (J):
First, calculate the fixed components:
- (1 / 7,304) * 500 ≈ 0.0685
- (1 / 178) * 15 ≈ 0.0842
- (1 / 13) * 1 ≈ 0.0769
Summing these:
0.0685 + 0.0842 + 0.0769 ≈ 0.2296
Now, the inequality becomes:
1 ≤ (1 / 1,533,939) * J + 0.2296
Subtract 0.2296 from both sides:
0.7704 ≤ (1 / 1,533,939) * J
Multiply both sides by 1,533,939 to solve for J:
J ≥ 0.7704 * 1,533,939 ≈ 1,181,000
Conclusion:
Therefore, the jackpot must be at least 1,181,000 for the expected return from playing Gopher 5 to be favorable to the player. This means that if the jackpot is below this amount, the expected value of purchasing a ticket is less than the cost of the ticket, making it an unfavorable bet.
Note: This calculation assumes there is only one winner and does not account for taxes or other potential deductions.