How many people in a room share a birthday?

The above is a standard probability question- there are different precise framings, one of which is 'how many people need to be in a room to have greater than 50% chance that at least two share a birthday' the answer being a surprisingly low 23.

So I thought I was on fairly safe grounds this morning when giving a talk at a primary school…

I never work on the 8th November. No tax, no clients, no Deloitte. Of course, that is not entirely true- I act for two national retailers whose shops are on every high street; and I listen to the business news, so can't help think of tax…but 8th November was the day my mother died, now eleven years ago, and I swore to myself I would never work on that day, day, and just do nice things. There are 364 other days in the year. Today has started really well. Some weeks ago I was asked by one of the teachers at my children's former primary schools if I would come to speak to their assembly, as one of their series of outside 'inspirational' speakers, and I was glad to accept: if a bit frightened of speaking in front of c 300 children, never having done so before. But it went really well, and I feel exhilarated having done it.

I has been told by the organising teacher that might be a good idea to ask the children questions, which I decided to do [Jane's sum total of her advice, despite her being a primary school teacher, was 'don't look scruffy': thank you, darling].

So, I asked them 'what is the capital of Iceland' (one person knew the answer, and then I asked them did anyone have a birthday on 9th December (no); 22nd July (no); 17th July (no). I explained to the children that I would come back to Reykjavik, and these dates later, as I did- they were respectively the date [1980] I beat Viktor Korchnoi, the date [2012] of the Allan Beardsworth 50th birthday tournament in Reykjavik, and the date [1976] that my school won the Sunday Times chess tournament – three significant dates.

Fortunately, it didn't matter- hands shot up with answers like 'mine is the 10th December' so it didn't matter one jot, but what is the probability of this happening: that no-one in audience of 300 have birthdates of and of three random dates?

I am not sure of my answer, but think it is (364/365)^300 * (363/365)^300 * (362/365)^300, or about 0.7%. In other words, there was a greater than 99% chance that at least one child would have had one of the three dates as a birthday.

Greater than 99%: but it didn't happen.

Postscript

I doubt my above maths- 99% just doesn't seem right, and asking a colleague, he thinks, probably correctly, that the true odds are 1- (362/365)^300, which is 92%. More likely.

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