Why No One Has To Win the Lottery

Posted on May 5, 2006
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Apparently tonight’s Ontario lottery is worth $35 million. That’s a lot of cash, enough for the media to do a piece on it in the news tonight. One common remark from everyone interviewed: “Someone’s gotta win it so why not me?”

Besides the shear improbability of anyone with a ticket winning (in this case greater than 1 in 13 000 000) each person interviewed subscribes to the same fallacy: that someone has to win.

In raffle ticket draws this is true and this is because the tickets sold make up the complete set of possible winning values. In a lottery such as Lotto 6/49 (or Powerball for the American readers) the complete set is defined by all the possible numbers that can be randomly selected by a computer, of which the numbers picked by people playing the lottery are a subset. This means that unless the number of unique combinations picked by people playing the lottery is equal to or greater than the number of possible numbers that can be selected it is possible that absolutely no one will win.

Consider it this way: a simple lottery in which the winning number is a single digit chosen from between 1 and 9 inclusive:

   1, 2, 3, 4, 5, 6, 7, 8, 9

Three tickets are sold: 2, 7 and 8.

If 1, 3, 4, 5, 6 or 9 are chosen then no one wins, everyone loses their money and the pot is increased for next week’s draw (now you know why the lottery jackpot increases each week until it suddenly drops back down to a pay-no-attention small value at some point).

In other words: in our fake draw above the odds of any person with a ticket winning are 1 in 9 (11%) and the odds of anyone winning it at all are 3 in 9 (33%). Not great odds no matter how you look at them.

Contrast that to a raffle draw in which these tickets are sold (assume one ticket per person):

   341, 342, 343, 344, 345

Since they define the possible options for winning are defined by those tickets the chances of any one person winning are 1 in 5 (20%) and the chances that anyone will win are 5 in 5 (aka: 100%).

So, play the lottery and split the winnings with me but know, unlike the people on the news, that there’s a good chance no one will win it and that the only one who truly wins with the lottery is the governemt. No matter what happens they always get their cash.

(Incidentally the numbers that you choose when playing the lottery are meaningless as is playing the same numbers each week, and though most people know that rationally they still do it superstitiously for some reason.

The only way to possibly increase your odds of winning is to buy more tickets - that is, increase the size of your own personal subset. Be prepared to buy a lot of tickets before they have any meaningful impact on the odds).

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