Eh. If I were planning to be alive for 56 choose 6 years or so, maybe I'd do that every time. But even then, probably not, since the expected number of winners would probably demolish the expected gain anyway.
The expected winnings are not greater than the cost of the ticket, because there's a truckload of tickets sold, and you split any prize you get with everyone else who picked the numbers...
Less people play means bigger portion of the payout. If 100 people win, I'm not getting "very much" of the $500 million, but if 1 person wins the following week, I'm getting more.
Small payout divided by smaller number of winners isn't necessarily larger than larger payout divided by larger number of winners. If 100 people split the 500m (which necessitates nearly 20b tickets sold), that's 5m each. The jackpot could be 4m the next week. The expected prize per winner depends on the ratio of the jackpot to the number of tickets sold.
The number of winners is proportional to the number of tickets sold (and the jackpot also increases), so unless they sold over 640m tickets, you're still coming out ahead.
Let n be the number of tickets purchased in this go-round. Let J denote the value of the jackpot, let W(n) denote the expected number of winners, and let's ignore the lower valued prizes.
The jackpot increases as a linear function of n, so J(n) = 640 million + cn, where 0 < c < 1. The probability of any given ticket being a winner is fixed: (56 choose 5)*(1/56)^5 * (1/46), and so W(n) = (56 choose 5)*(1/56)^5 * (1/46) n.
Suppose everyone else is locked out of the lottery for some reason, so you have the lottery entirely to yourself, and you purchase exactly one ticket for each combination. That's about 176 million tickets, so the payout is 464 million, less labor costs. So if you can keep other people from playing, and you have 176 million dollars lying around, and you can hire a few thousand people to fill out tickets for a week, it's a good deal, paying out $3.64 per dollar invested
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There's also the utility function to consider.
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The number of winners is proportional to the number of tickets sold (and the jackpot also increases), so unless they sold over 640m tickets, you're still coming out ahead.
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The jackpot increases as a linear function of n, so J(n) = 640 million + cn, where 0 < c < 1. The probability of any given ticket being a winner is fixed: (56 choose 5)*(1/56)^5 * (1/46), and so W(n) = (56 choose 5)*(1/56)^5 * (1/46) n.
Suppose everyone else is locked out of the lottery for some reason, so you have the lottery entirely to yourself, and you purchase exactly one ticket for each combination. That's about 176 million tickets, so the payout is 464 million, less labor costs. So if you can keep other people from playing, and you have 176 million dollars lying around, and you can hire a few thousand people to fill out tickets for a week, it's a good deal, paying out $3.64 per dollar invested ( ... )
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...I just read that nearly 1.5 billion tickets were sold. Was not expecting that.
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