A simple, silly, but entertaining example of mathematical illiteracy by way of the Associated Press:
OMAHA, Neb. (AP) -- The odds are against something this odd. But a Nebraska Lottery official says there was no mistake: The same three numbers in Nebraska's Pick 3 lottery were drawn two nights in a row this week.
Lottery spokesman Brian Rockey said one of two lottery computers that randomly generate numbers produced the numbers 1, 9 and 6 -- in that order -- for Monday night's Pick 3 drawing. Rockey says the next night, the lottery's other computer produced the same three numbers in the same sequence.
The odds of such an occurrence? One in a million.
Close... Only off by three orders of magnitude...
Assuming a fair system, and assuming that the drawing system
can produce the same number in multiple positions, the odds of drawing the numbers "1,9,6" twice in a row are, indeed, one in one million. But the odds of drawing the same number two nights in a row are just one in 1/1000.
An easy way to think of it: the first days draw doesn't matter. Whatever number it produces is unimportant - any value for the first draw results is fine. The question is, what are the odds of the second draw producing the same result as the first?
Three digits means 1000 possibilities - so 1 in 1000.
Of course, it looks like they got even more than that wrong. From what I can find in a quick search, the Nebraska lottery draw produces three
different digits - once a number has been drawn, it can't be drawn again. So you'll never get a result like "998", because you can only draw one 9. So in fact, the total number of possible draws isn't 1000. It's 720. So the odds of a duplicate draw on a given pair of nights is just 1 in 720.
And, of course, if you really wanted to figure out how likely it was for this to happen, you wouldn't choose one particular pair of nights. The real question of probability here isn't "How likely is it that tonight and tomorrow, a three-draw lottery will produce the same draw results?" The real question is, "If I operate a three-draw lottery for a year, what are the odds that I'll draw the same result two nights in a row?". And the answer to that is: not terribly unlikely.
It's like the old birthday game: in a class of 30 people, what are the odds that two of them will have the same birthday? According to the reasoning
of the AP, it would be something like 1 in 133,000. In reality, it's around 2 in 3.