rare-events

In Why coincidences happen we saw how the chance of a rare event occurring to someone, somewhere, depended on the number of opportunities for it to occur.

In What are the chances? we saw how the chance of a rare event occurring could be calculated for specific problems.

When we experience a surprising event and wonder about the likelihood of such a coincidence, we may be able to use probability theory to work out the chance of it happening. And whether the coincidence happens to us or to someone else, we need to take into account how many opportunities there are for it to happen.

We recently heard about a couple who have two children of
different ages, but who were both born on the same date, and in exactly the same minute - say 7.32 am on September 30th. What is the chance of this happening to them, or to any family in this country?

Consider a lottery in which there are $N$ possible number combinations - in the UK Lottery $N $= 13,983,816. Each ticket therefore has a $1/N$ chance of sharing in the jackpot. Suppose we sell $N$ tickets, what is the chance of nobody winning the jackpot? What if we sell $2N$? The chances are shown in the table below, and hold whatever $N$ is, provided it is large!

A recent news story featured a family whose three children had all been born on January 29th. But is this so remarkable?

You bump into an old friend you haven't seen for years.
You find that you share a birthday with not just one but two other people in your office.
You win the lottery!