You've probably heard that buying a lottery ticket is a huge waste of money. People call it the "idiot tax."
Normally, that's harsh but relatively accurate.
The jackpot has to be at least $275 million for it to be considered remotely worthwhile.
But tonight's Powerball lotto is the exception to that rule.
Because of the high stakes, the expected winnings of buying a $2 ticket is $1.21. In other words, if you spend $2 on tickets, then you will get back $3.21 in winnings on average, for a net gain of $1.21.
Because Powerball publishes the probabilities of winning each prize, we're able to figure out the expected value of a ticket.
Slightly more than 96% of the time, a ticket is a loser, and a player loses $2.
But the remaining 4% of the time, the player enjoys a net financial gain. 2.7% of the time a player gains $2; 0.45% of the time a player gains $5; and so on and so forth. A tiny, tiny fraction of the time, a player gains half a billion dollars.
As a result, if you multiply through the probabilities you get the expected gains and losses of buying a ticket. The average gains of buying a ticket exceed the guaranteed loss of buying a ticket because of the huge jackpot, and the net expected value of playing a ticket turns out to be $1.21.
What's more, there's probably going to be a winner this time.
Here's the probability of at least one jackpot winner given the number of tickets sold:
Powerball claims that they expect to sell more than 189 million tickets before the drawing tonight. That translates to a 66% probability of at least one winner, so the odds are rather good that there will be at least one new multi-millionaire tomorrow.
The best way to determine that figure out is 1 minus the probability that there isn't a winner at all, that every single ticket sold is wrong. Since the probability that a single ticket is wrong is 175223509/175223510, that raised to the 189 millionth power is around 34 percent. There's a 34% probability that every ticket is wrong, so there's a 66% probability that at least one ticket is right.
This brings us to the conclusion that there is a two in three chance that the Powerball lottery won't get any better than it is now.
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