Alex's Adventures in Numberland - Alex Bellos [137]
If a group of 23 people is not immediately accessible to test this out, just look at your immediate family. With 4 people it is 70 percent likely that 2 will have birthdays within the same month. You y need 7 people for it to be likely that 2 of them were born in the same week, and in a group of 14 it is as likely as not that 2 people were born within a day of each other. As group size gets bigger, the probability rises surprisingly fast. In a group of 35 people, the chance of a shared birthday is 85 percent, and with a group of 60 the chance is more than 99 percent.
Here’s a different question about birthdays with an answer as counter-intuitive as the birthday paradox: how many people do there need to be in a group for there to be a more than 50 percent chance that someone shares your birthday. This is different from the birthday paradox because we are specifying a date. In the birthday paradox we are not bothered who shares a birthday with whom; we just want a shared birthday. Our new question can be rephrased as: given a fixed date, how many times do we need to roll our 365-sided dice until it lands on this date? The answer is 253 times! In other words, you would need to assemble a group of 253 people just to be more sure than not that one of them shares your birthday. This seems absurdly large – it is well over halfway between one and 365. Yet randomness is doing its clustering thing again – the group needs to be that size because the birthdays of its members are not falling in an orderly way. Among those 253 people there will be many people who double up on birthdays that are not yours, and you need to take them into account.
A lesson of the birthday paradox is that coincidences are more common than you think. In German Lotto, like the UK National Lottery, each combination of numbers has a 1-in-14-million chance of winning. Yet in 1995 and in 1986 identical combinations won: 15-25-27-30-42-48. Was this an amazing coincidence? Not especially, as it happens. Between the two occurrences of the winning combination there were 3016 lottery draws. The calculation to find how many times the draw should pick the same combination is equivalent to calculating the chances of two people sharing the same birthday in a group of 3016 people with there being 14 million possible birthdays. The probability works out to be 0.28. In other words, there was more than a 25 percent chance that two winning combinations would be identical over that period; so the ‘coincidence’ was therefore not an extremely weird occurrence.
More disturbingly, a misunderstanding of coincidence has resulted in several miscarriages of justice. In one famous California case, from 1964, witnesses to a mugging reported seeing a blonde with a ponytail, a black man with a beard and a yellow getaway car. A couple fitting this description were arrested and charged. The prosecutor calculated the chance of such a couple existing by multiplying the probabilities of the occurrence of each of detail together: for a yellow car, for a blonde, and so on. The prosecutor calculated that the chance of such a couple existing was 1 in 12 million. In other words, for every 12 million people, only one couple on average would fit the exact description. The chances of the arrested couple being the guilty couple, he argued, were overwhelming. The couple was convicted.
The prosecutor, however, was doing the wrong calculation. He had worked out the chance of randomly selecting a couple that matched the witness profiles. The relevant question should have been, given there is a couple that matches the description, what is the chance that the arrested couple is the guilty couple? This probability was only about 40 percent. More likely than not, therefore, the fact that the arrested couple fitted the description was a coincidence. In 1968 the California Supreme Court reversed the conviction.
Returning to the world of gambling, in another lottery