The Lottery Paradox
Suppose you hold one ticket ( 
, for
some 
) in a fair lottery consisting
of 1 million tickets, and suppose it is known that one and only one ticket
will win. Since the probability is only .000001 of 's being drawn,
it seems reasonable to believe that will not win. By the same
reasoning it seems reasonable to believe that 
will not win,
that 
will not win, 
, that 
will not win.
Therefore it is reasonable to believe
But we know that
So we have an outright contradiction.
Selmer points out that the reasoning here is precisely what led he and his wife to an agreement that they not play the lottery. Selmer reasoned like this:
``Look, the odds of a door falling off a 747 and hitting me on the head when I go outside are rather slim -- so much so that I don't bother looking up when I do venture out. Similarly, I don't bother checking to see if I've won the lottery: the odds against it happening are such as to make it irrational to look."
His wife objected:
``That can't be right! After all, we could win!"
And Ryan Hope sends in an anecdote that may shed some light:
I know a man named Joe from Allentown, PA who used to play the lottery all the time. He was an office supply salesman. He would buy several lottery tickets every day. Sometimes he would look at them, and sometimes not. But he would always file the tickets in file folders by date. You should also know that, at least in Pennsylvania, lottery tickets can be redeemed for up to 1 year after which they are not valid any more.One Saturday, Joe was reading the paper. There was a small article urging people to check their lottery tickets. There was an unclaimed jackpot that was to expire on the following Monday. So Joe went to his neatly filed lottery tickets to check if he held the winning ticket. Sure enough he had this ticket in his folder. Imagine his joy when he realized that he had won 20.9 million dollars, to be paid out in installments of $990,000 per year for 26 years. And if he had waited just 2 more days to check his tickets, he would have lost it all.
Ryan Hope
Lockheed Martin, VF