Math is Hard, Part Deux

George Mason University Professor Alex Tabarrok on the expected value of playing tonight's Powerball:
The instant payout is about $496 million so that makes the expected value 496*1/292.2=$1.70. We also have to adjust for the possibility that more than one person wins the prize. [...] with so many people playing it wouldn’t be surprising if two people had the same number–I give it at least 25%. So that knocks your winnings down to $372 million in expectation.
In a word, no.

Approximately 300 million tickets have been sold for this draw, and there are 292 million possible combinations. That means that on average, at least one winning ticket has already been sold (could be zero, could be two or three, but one is the most likely outcome).

Tabarrok's error is that he is looking at the likelihood of two winners in general. Which is a very different question than "Assuming I have the winning combination, what is the likelihood that zero of the other 300 million tickets are winners?" The two events are independent; my having the winning numbers does not reduce the chances of the other 300 million tickets winning. The odds that if you won, you'd have to share the prize are well over 50%, not 25%.

For related ruminations on the other big multi-state lottery game, Mega Millions, see here.

1 comment:

Garfield Bertie said...

I agree with you as for the calculations. I have been playing and checking USA lottery results for almost 10 years already playing several games simultaneously. As for the chances of winning such huge lottery as Mega Millions or Powerball are worse even opposed to the chances to be born a millionaire in the USA.

Happy Super Tuesday!