A birthday connundrum
It’s nice to know that I am not the only one to get dates wrong. Yesterday I received a couple of “Birthday Greetings” … a day early, but it’s also nice to know that people are thinking about me.
Last year I did the same … and sent my nephew a congratulatory message on Facebook. He thanked me and politely put me in my place by telling me that I was a day early. I looked in my diary again, and then at his Facebook profile for confirmation, and to my embarrassment had to admit that, sure enough, I had made a mistake. If only I had waited for Facebook to “remind” me!
Actually, it should be fairly easy to remember my birthday, especially here in Kyrgyzstan, simply because it is a public holiday. I like to pretend that this is a sign of respect offered to me by Kyrgyzstan, but it goes much further back than that … It is International Women’s Day and this year, depending on your sources and how you calculate these things, marks the hundredth such celebration. (I wrote a postcard about the history of IWD, International Women’s Day, two years ago: http://ianbek.kg/?p=827).
As a result of the public holiday, we tend to have mark my birthday early in the office … everybody gathered and we had a cake and a little presentation. It was all very pleasant and very light hearted.
John was telling me that he has a much easier time remembering birthdays in his family. Most of them fall on dates like February 2nd, March 3rd, April 4th, and so on, (2/2, 3/3, 4/4 …). When he told me I was surprised and asked him if he had ever worked out the probability of such a thing happening. Even though the family has a number of mathematicians in it, I don’t think that the thought had ever occurred to anyone.
I was reminded of something I read in a Martin Gardner book on recreational mathematics … no, that’s NOT a contradiction of terms. He was writing about statistical anomalies and posed the question how many people do you need to be certain that at least two of them share the same birthday. Now that’s easy, 367. Just in case everyone in your selected group was born on a different day, then to make sure that at least two people shared a birthday, you need one for everyday of the year, (don’t forget February 29th), plus one: 367.
But, he went on, you can be fairly sure with a much smaller number.
He suggested that you get such a group of people together and get them to write down their birthdays on pieces of paper, (to avoid any cheating), and then collecting them together to go through. If the group is small, say just 8 or 9 people, then the probability is pretty high that they will all have different birthdays, but as the size of the group grows so the probability grows that at least two of them will have written down the same date.
What’s more, it doesn’t matter if someone realizes what is going on and chooses to lie, writing down a totally different date. The probability is such that they will have chosen one of the dates written down by someone else – unless, of course, there is some collusion.
If that sounds incredible, don’t worry, I found it difficult to accept as well, even though he went on to give an explanation. Anyway, I decided to try it out. I went into one of my classes which happened to doing Probability at the time, and said that we were going to conduct an experiment. Fortunately, it was a fairly big class – 30 kids. I knew, thanks to Martin Gardner, that once the group got to 23 or 24, it was already a fifty-fifty chance and every additional person just improved the odds in my favour.
I gave out the pieces of paper and instructed the children to write down their birthdays – just the day and the month – collected them in, explained what we were going to do. The kids didn’t believe it either and I made a rash offer … if all the dates were different then I would treat all of them to a Mars Bar, and they agreed that if two of them were the same then they would each buy me a Mars Bar. What did they have to loose? If they lost, then it would cost each one of them something like 15 pence, but if I lost, then it would cost me almost a fiver, (five pounds).
So, we set about examining the slips of paper and writing the dates up on the blackboard. As the first few dates were inspected there were few surprises, they were well and truly scattered throughout the year. I wasn’t too worried, after all, I had Martin Gardner’s assurance that the odds were on my side. I started to get worried when we got about half way, fifteen dates up on the board and all them different. The kids, of course were enjoying this immensely, and smacking their lips in anticipation … until date number 21 was drawn from the pile of slips … phew, it was already up there on the board. Groans all round from the kids, and a sigh of relief from me.
I told them that I wouldn’t hold them to the “bet”, and that they should learn the lesson that I call the Sky Masterson lesson. In the film “Guys and Dolls”, Marlon Brando in the Masterson role explains to Frank Sinatra and Chubby Kaye, that his father warned him never to accept a bet if someone comes in and offers a bet that seems to a certainty, that they can’t lose. For example, if they want to bet that they can make the drink sitting on the table in front of him to jump up and empty itself all over his head, and he takes the bet, then he is sure to end up with a wet head and minus the stake.
After all, I pointed out, I could have cheated by checking first of all with the class register. In fact, I hadn’t, but I am not sure that all of them were convinced of this.
So, I then set about the explanation. Part of the reason why it seems to be an incredible coincidence and counter intuitive that two people should share the same birthday, given that the group was so small if you remember that you need 367 people to be certain of the result, is that people start by asking the wrong question: what is the probability that two people will share the same birthday?
Given just two people, then the probability that they share a common birthday is 1/365 x 1/365, ( … let’s forget February 29th as that makes the numbers even bigger … or smaller …), which is 1/13325, which is an incredibly small number, (about 0.000007506). So small, in fact, that you can probably discount it as so unlikely as to be practically impossible. Intuition seems to tell us that even by introducing other people, the odds are still going to be incredibly slight, even if we are not sure about how to go about making the calculation. One person, (believe it or not a Maths teacher), when faced with the question told me that the intuitive answer was 50-50 … it either happened or it didn’t.
The right way to tackle the problem, however, is to ask a different question, but one which has the same meaning: What is the probability that none of them share a common birthday?
Given two people, the chances that they were born on different days of the year is quite high: 364/365, or 0.9973. That’s almost tantamount to a certainty. There’s a slight risk, (about one quarter of one percent), that they will share the same birthday, but it’s so insignificant that it is almost negligible.
Introduce a third person, however, and the odds that they will have a different birthday is only 363/365, or 0.9945, and the probability of all three having different birthdays is 364/365 times 363/365, or 0.9836. That’s still pretty “certain” … but less “certain”, (if that’s not a contradiction in terms).
Add a fourth person and the probability falls … to 0.9729, and when you add a fifth, it falls again, to 0.9595.
By the time you have gathered 23 people together, then the odds are already 50-50 that they will all have all have different birthdays, or that at least two of them will share common birthday. Think of it this way, with twenty three people on a football pitch, (that’s soccer for the Americans … two teams of eleven plus the referee), then there is a fifty-fifty chance that two of them will share a birthday, (which should have pleased my colleague).
With a class of thirty, the probability that they will all have different birthdays has fallen to 0.2937. In other words, the possibility that at least two of them will share a birthday has risen to 0.7063. Not a certainty by any means, but a considerable improvement.
In terms of statistical analysis, you need about 41 people to be 90% confident of “success”. Only another 5 people gives you 95% confidence, which is pretty impressive and the sort of level used in statistics as an indicator that the results are “significant”, and once you get to 57 people, you can be 99% confident that you will have two people sharing a birthday. By no means a certainty, but you could conduct the experiment 100 and only expect to lose the bet on one occasion.
That’s not bad odds.
Nothing in life is simple, however, and the situation gets more complicated if you specify the date, (for example, what are the odds that someone will, like me, have a birthday that falls on a public holiday), or if it turns out that birthdays are not spread out evenly throughout the year, which is by no means certain. I’m not sure how to work out the probabilities in John’s case, but it provided me with food for thought.
.
As it happens, I am not too fussed about celebrating my birthday. Although people here make a lot of fuss celebrating Birthdays, I probably couldn’t care less about it. To me, it’s just another day … another working day, even if it is a public holiday. That’s one of the drawbacks of the hotel business … like the Windmill, “We never close!”
Bishkek