Many lottery games for very large prizes include guessing which numbers will appear from a random sample of the numbered balls

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.

citeturn0search0

Calculating the Expected Value (EV):

The EV is calculated by multiplying the probability of each outcome by its respective prize and summing these products:

1. Jackpot Prize:

• Probability: 1 / 1,533,939
• Prize: $J (the jackpot amount)
• Contribution to EV: (1 / 1,533,939) * J

1. Match 4 of 5:

• Probability: 1 / 7,304
• Prize: $500
• Contribution to EV: (1 / 7,304) * 500

1. Match 3 of 5:

• Probability: 1 / 178
• Prize: $15
• Contribution to EV: (1 / 178) * 15

1. 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.