Question

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 $1, 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.

Answer 1

The jackpot amount needed for a favorable return is $1,543,940.



Let's perform the calculations step by step:

1. Calculate the probabilities of winning each prize:

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`\( P(\text{win jackpot}) = \frac{1}{\text{odds jackpot}} \)`

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`\( P(\text{win 5 of 5}) = \frac{1}{\text{odds 5 of 5}} \)`

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`\( P(\text{win 4 of 5}) = \frac{1}{\text{odds 4 of 5}} \)`

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`\( P(\text{win 3 of 5}) = \frac{1}{\text{odds 3 of 5}} \)`

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`\( P(\text{win 2 of 5}) = \frac{1}{\text{odds 2 of 5}} \)`

2. Substitute the values into the expected return formula:

`\[ \text{EV} = P(\text{win jackpot}) * \text{payout jackpot} + P(\text{win 5 of 5}) * \text{payout 5 of 5} + P(\text{win 4 of 5}) * \text{payout 4 of 5} + P(\text{win 3 of 5}) * \text{payout 3 of 5} + P(\text{win 2 of 5}) * \text{payout 2 of 5} - \text{Cost of playing} \]`

3. Set the expected return equal to zero and solve for the jackpot amount.

Let's perform these steps:

1. Calculate the probabilities:

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`\( P(\text{win jackpot}) = (1)/((1)/(1,533,939)) \)`

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`\( P(\text{win 5 of 5}) = (1)/((1)/(13)) \)`

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`\( P(\text{win 4 of 5}) = (1)/((1)/(178)) \)`

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`\( P(\text{win 3 of 5}) = (1)/((1)/(7,304)) \)`

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`\( P(\text{win 2 of 5}) = (1)/((1)/(15)) \)`

2. Substitute the probabilities into the expected return formula:

`\[ \text{EV} = (1)/((1)/(1,533,939)) * 1 + (1)/((1)/(13)) * 1 + (1)/((1)/(178)) * 15 + (1)/((1)/(7,304)) * 1 + (1)/((1)/(15)) * 1 - 1 \]`

3. Simplify the expression to find the jackpot amount.

Let's perform the calculations:

`\[ \text{EV} = 1,533,939 + 13 + 15 * 178 + 7,304 + 15 - 1 \]`

`\[ \text{EV} = 1,533,939 + 13 + 2,670 + 7,304 + 15 - 1 \]`

`\[ \text{EV} = 1,543,940 \]`

Therefore, the jackpot amount needed for a favorable return is $1,543,940.


The probable question can be: 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