Why coincidences happen

birthday cakeWhen 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.

Suppose you stop a stranger in the street and ask them their birthday. If they reply politely, and you find they have the same birthday as yourself, you would be quite surprised. But consider the following conversation:

penguins

When is your birthday then?

Fred: "I met Tony today and he had the same birthday as me!"
You: "Amazing! (pause) How many people did you meet today?"
Fred: "Oh, about 1000"

We are now distinctly unimpressed by Fred's story.

There are four stages to analysing coincidences:

  1. Work out the chance of the specific event occurring. For example, there's a 1 in 365 chance of Fred having the same birthday as Tony.
  2. Work out how many opportunities there were for a similar event to occur. Here Fred had 1000 opportunities to meet someone with the same birthday.
  3. Multiply the specific chance by the number of opportunities to get what is known as the expected number of events.
    In this case the expected number is $1/365 \times 1000 \approx 3$, so we would expect Fred to have met around 3 people with the same birthday.
  4. Work out the chance of observing at least one of these events. These are shown in the Table below. As Fred expected to meet 3 people with the same birthday, the chance that he didn't meet anyone with the same birthday is around 5%. So we would be quite surprised if he said he did not meet anyone out of those 1000 people who shared his birthday!
Table showing how the expected number of events tells you the chance of no event occurring
Expected number of events Chance no events occur Chance at least one event occurs
$1/2$ 61% 39%
$1$ 37% 63%
$2$ 13% 87%
$3$ 5% 95%
$4$ 2% 98%
$5$ 1% 99%

See how this applies to the chance of someone, somewhere, winning the jackpot in the National Lottery.

In What are the chances? we show how this works in some specific examples, and in we provide a more mathematical treatment in Maths of coincidence.

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