For a better chance of a match, say 95%, we need to approximately double this number to 2.5 √C. So if we have N = 2.5 √365 = 48 people in a room, it is very likely indeed that two will have the same birthday.
This makes it easy to make money from people. Suppose you have 30 people together. Bet the group that two of them have a birthday within one day of each other. What are the chances you will win?
First consider the chance that any two people (say me and you) match in this way: if my birthday is August 16th (which it is), then a match would happen if you were born on the 15th, 16th, or 17th, which is 3 out of 365 days, or a 1 in 122 chance. So C = 122 in this case. This means that for a 50% chance of a match we only need 1.2 √122 = 13 people, and for a 95% chance we need 2.5 √122 = 28 people. In other words, with 30 people in a room you are almost certain to win.
Regardless of the number of people gathered together, you can make money off them provided they are a bit gullible, preferably drunk, and not good at probability. Suppose there are N = 50 people: and say we reverse the 95% chance equation N = 2.5 √C to give C = (N/2.5)2. This means that when N = 50, then C = 20 x 20 = 400.
So, get these 50 people to choose a number at random between 1 and 400, and bet them that they will not all choose different numbers. Even if they choose completely at random, there is a 95% chance there will be a match. And people tend to choose particular numbers anyway – avoiding those ending in a zero, preferring odd numbers and so on – increasing the chance of match. However, although you may make money, you may also lose friends.
The final explanation for coincidences is what is called the law of truly large numbers, which says that anything remotely possible will eventually happen, if we wait long enough. Or to put it another way, even genuinely rare events will occur, given enough possibilities.
For any three people, say children in a family, there is a 1/365 x 1/365 = 1 in 135,000 chance of them all sharing the same birthday, and even more if there is some planning going on. This is clearly a rare event. But there are one million families in the UK with three children under 18, and so we should expect around eight families to have children with matching birthdays, and that new cases crop up around once a year. Which they do: new examples in the UK occurred on 29 January 2008, 5 February 2010 and 7 October 2010.
So given all this, it would be really strange if memorable coincidences did not happen to you. But this may be difficult to keep in mind when you are walking past a phone box, it rings, you decide to answer it, and you find the call is for you. When this happens to someone, they remember it for years.
But just think of all the people you have ever known. Then think of all the people that you have had some connection with, such as attending the same school, being friends of friends and so on. It will be tens of thousands. If you are the sort of person who talks to strangers, you will keep on finding connections. If you are not, then think: you might have sat on a train next to a long-lost family member, and never realized it.