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I have $\$250$ million in the bank. The odds of winning the lottery jackpot are about $1$ in $250$ million. The lottery payout gets up to $\$300$ million. A ticket costs $\$1$. The jackpot is the only payout. I am the only person playing the lottery. The lottery people can instantly tell me if I win or not, and I don't have to look through my tickets. If I buy tickets, they will be randomly selected. Duplicates are a possibility.

Should I spend $\$250$ million on lottery tickets? What is the expected value of the gamble? What is the percentage chance that I would win?

How the lottery works: you pick a set of random numbers. The lottery people pick a set of random numbers. The chances those random numbers will be identical are 1 in 250 million. If any of your tickets math the lottery numbers, you receive $\$300$ million, but not your original $\$250$ million spent. If none of your tickets match, you win nothing, and lose your $\$250$ million wager.

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    $\begingroup$ If you are risk averse, I'm sure you wouldn't. Maybe you should rephrase your question (I think you really mean to ask about the expected value here). $\endgroup$ – Tyler Dec 19 '14 at 19:34
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    $\begingroup$ "up to $300 million" --- more specifics, please. Does this mean it might be $300 million, but it also might be zero? You see how this might affect your answer? $\endgroup$ – Michael Grant Dec 19 '14 at 19:39
  • $\begingroup$ @MichaelGrant the jackpot is the only payout. Could it also be zero is part of the question. General knowledge of how the lottery works is assumed. $\endgroup$ – Evorlor Dec 19 '14 at 19:41
  • $\begingroup$ more specifics: Do you choose the ticket numbers yourself, or are they chosen randomly? (And if randomly, with or without replacement?) Are duplicate tickets possible (e.g., more than one winner)? $\endgroup$ – Michael Grant Dec 19 '14 at 19:41
  • $\begingroup$ @MichaelGrant good point. Updated. (Random yes duplicates yes) $\endgroup$ – Evorlor Dec 19 '14 at 19:43
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Well, I'm not dispensing financial advice, so it all depends on how you look at it.

The expected value of each dollar spent on a lottery ticket is $\dfrac{300,000,000}{250,000,000}=\$1.20$. Therefore, you could spend money on the lottery and expect to make a profit. Looks good!

However, assuming the tickets chances of winning are independent of each other, the number of tickets you need to purchase before winning is geometrically distributed with $p=1/250,000,000$. The probability that this number is less than or equal to $250$ million is given by the Cumulative Distribution Function for this geometric variable: $$p(x<250,000,000)=1-\left(1-\frac{1}{250,000,000}\right)^{250,000,000} \approx1-\frac{1}{e}\approx0.63$$ So with your $\$250$ million you have about a $63\%$ chance of winning the jackpot--which gives an expected outcome of $0.63\cdot300,000,000=\$189$ million. Maybe it isn't such a good idea after all.

There is one other thing to consider, though--the outcomes may not be independent. If the lottery uses the common method of selecting a sequence of numbers, presumably you would not buy a ticket with numbers that you already knew were wrong. In this case, there are only $250$ million possible ticket numbers, so if you buy one of each, you're guaranteed to win!

Edit:

This answer was written before the question was revised and clarified, which negates my final paragraph. The expected value on the first part assumes that you can win the jackpot more than once, the second part assumes that you cannot. It also assumes that you will spend the entire $\$250$ million no matter what. If you can buy your tickets one at a time and stop when (if) you win, that changes the outcome and the calculations become more complicated. However, there is still a $\sim37\%$ chance that you get nothing, so I would guess that, while a better deal than otherwise, it still wouldn't be worth it.

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  • $\begingroup$ I guess I'll go buy an island instead! Thanks! $\endgroup$ – Evorlor Dec 19 '14 at 19:55

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