Recently I came across an interesting idea about little mistakes in counting problems that actually don’t amount to much. In A Problem Squared 030, Matt Parker was investigating the question “What are the odds of having the same child twice?” and made some simplifying assumptions when thinking about DNA combinatorics. He justified leaving out a small number of things when counting an astronomical number of things by going through an example from the lottery.
The current UK lottery uses 59 balls and draws 6 of these, so the one in 45 million figure arises from \(\binom{59}{6}=45,\!057,\!474\), and the probability of winning is a tiny
\[ \frac{1}{45057474} = 0.00000002219387620 \text{.}\]
Matt posits the idea that somewhere along the way we forget to include some tickets.
But let’s say along the way while I’m working it out, for strange reasons I go ‘oh you know what, I’m going to ignore all the options which are all square numbers. You know, I just can’t be bothered including them. Yeah, they’re legitimate lottery tickets, but just to make the maths easier I’m going to ignore them’. And people are getting up in arms, and they’re like ‘you can’t ignore them, they’re real options’.
