NOOKS AND CRANNIES



How many people do you need to gather together in order to make it probable that two of them will share the same birthday? I am sure that I once read that it is less than thirty. Denis Purshouse, London England When 23 people are gathered, there is more chance than not that 2 of them have the same birthday. In order to help understand this, you should consider how many PAIRS of people there are. With 23 people there are 253 different pairings i.e. more than half of the 365 possible birthdays. Paul Wright, Basildon UK 23 people have a slightly over 50% probability of two of them sharing the same birthday. Robert Hanstock, Pangbourne England It depends what you mean by probable. Presuming that you mean better than a 50% chance then the answer is 23 people and the reasoning (ignoging leap years) goes :- Start with one person in a room. Add a second person and the probability that he has a different birthday is 364/365. Add a third and the probability that he has a different birthday to the first two is 363/365. The probability that there are now no duplicate birthdays is (364/365)*(363/365). Add a fourth and the probability that he has a different birthday to the first three is 362/365. The probability that there are now no duplicate birthdays is (364/365)*(363/365)*(362/365). etc. The probability that there are no duplicate birthdays falls below 50% when the 23rd person arrives. Tony Foxcroft, Gloucester UK People are very easily misled by statistics. Indeed it is said that when they are used in court cases, even the judges draw the wrong conclusions. It is common to hear that there is only one in a million chance of the DNA in the blood found at the scene of a murder matching that of the accused. So when there is a match, people conclude that the odds are a million to one that he did it. Whereas all we can really say is that it is possible he or she did it. There is no such thing as proof positive, one can only prove a negative. So turning to our birthday question. If we were to invite people one-by-one to come into a room, ignoring leap years, it is possible to gather together 365 people all with a different birthday. Taking people at random in this way, one would have to select 366 people to be absolutely sure that two of them had the same birthday. At the same time, it's fairly clear that it would be excruciatingly difficult to select 365 people from a crowd and not have two people with the same birthday. If you select just two people, the probability is very low, but it is possible! The more people you select, the more likely it is to pick two born on the same day. The magic figure of 20 comes in - in fact it's 23 - when the probability of having two people with the same birthday reaches 50/50. Put another way, if we selected many groups of 23 people we could be sure that half of the groups selected would contain two people with the same birthday. But it is never certain until you select 366. However when you get to 50, the odds amount to over 90%. Terence Hollingworth, Blagnac France 366, you would think. Steve, Bristol UK For two people, the probability is 1/365. Three people represents three combinations of two people, so the probability is 1/365 + 1/365 + 1/365 = 1/122. Four people gives 6 two-person combinations = 6/365 = 1/61. In order to be probable, ie. have a probability of 1/2 or greater, we need a number whose combinations add up to 365/2 = 182. Twenty people gives 190 two-person combinations, giving a probability that two of those twenty people share a birthday of just over 50%. Mark Lewney, Cardiff EU It's 23. Tim Ault, London UK The probability of two people in a randomly selected group sharing a birthday becomes greater than 50% when the size of the group reaches 23. John Charnock, Warrington UK 23! Students of Illuminati conspiracy theories will no doubt be intrigued by this. Nick, Milton Keynes UK Again we have contradictory answers. To see why Mark Lewney is mistaken, apply his method to a group of 28 people. There are 378 possible two-person combinations, so his method gives a probability greater than 1, which is clearly impossible. Applying the correct method as stated by Tony Foxcroft, the probability of a match for 20 people is just over 41 percent. Incidentally, for 28 people, the probability is just over 65 percent. This all ignores leap years and the fact that birthdays are not evenly distributed. For a given number of people, including leap years lowers the probability of a match slightly (for 23 people, from 50.73% to 50.69%). Taking account of the uneven distribution of birthdays (and the possibility that two of the people in the room are twins) increases the probability, but one would need full population data (which I do not have) to do the calculation. Pelham Barton, Birmingham UK The psychologist Susan Blackmore, when studying belief in parapsychology. She used this question as a way to test people's ability to estimate probabilities accurately. She found that people who were bad at estimating probabilities (i.e. gave answers a lot higher than 23) were much more likely to believe in telepathy, etc. Presumably because they underestimate the odds of 'strange' coincidences happening. Psychologists will obviously have to come up with a new question to test the odds-estimating powers of N & Q readers now... Sophia, Nottingham UK Here's a variation on the question: all things being equal, how many people tip the likelihood that two people within that group have a first name and last name that derive the same two initials? Thanks for any consideration. Non-techie Talk, Toronto Canada



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