Alex's Adventures in Numberland - Alex Bellos [136]
Randomness is not smooth. It creates areas of empty space and areas of overlap.
Random dots; non-random dots.
Randomness can explain why some small villages have higher than normal rates of birth defects, why certain roads have more accidents, and why in some games basketball players seem to score every free throw. It’s also why in seven of the last ten World Cup finals at least two players shared birthdays:
While at first this seems like an amazing series of coincidences, the list is actually mathematically unsurprising because whenever you have a randomly selected group of 23 people (such as two football teams and a referee), it is more likely than not that two people in it will share the same birthday. The phenomenon is known as the ‘birthday paradox’. There is nothing self-contradictory about the result, but it does fly in the face of common sense – twenty-three seems like an absurdly small number.
Proof of the birthday paradox is similar to the proofs we used at the beginning of this chapter for rolling certain combinations of dice. In fact, we could rephrase the birthday paradox as the statement that for a 365-sided dice, after 23 throws it will be more likely than not that the dice will have landed on the same side twice.
Step 1: The probability of two people sharing the same birthday in a group is 1 minus the probability of no one sharing the same birthday.
Step 2: The probability of no one sharing the same birthday in a group of two people is . This is because the first person can be born on any day (365 choices out of 365) and the second can be born on any day apart from the day the first one is (364 choices out of 365). For convenience, we will ignore the extra day in a leap year.
Step 3: The probability of no one sharing the same birthday in a group of three people is . With four people it becomes , and so on. When you multiply this out the result gets smaller and smaller. When the group contains 23 people, it finally shrinks to below 0.5 (the exact number is 0.493).
Step 4: If the probability of no one sharing the same birthday in a group is less than 0.5, the probability of at least two people sharing the same birthday is more than 0.5 (from Step 1). So it is more likely than not that in a group of 23 people two will have been born on the same day.
Football matches provide the perfect sample group to see if the facts fit the theory because there are always 23 people on the pitch. Looking at World Cup finals, however, the birthday paradox works a little too well. The probability of two people having the same birthday in a group of 23 is 0.507 or just over 50 percent. Yet with seven out of ten positives (even excluding the van de Kerkhof twins), we have achieved a 70 percent strike rate.
Part of this is the law of large numbers. If I analysed every match in every World Cup, I can be very confident that the result would be closer to 50.7 percent. Yet there is another variable. Are the birthdays of footballers equally distributed throughout the year? Probably not. Research shows that footballers are more likely to be born at certain times of the year – favouring those born just after the school year cut-off point, since they will be the oldest and largest in their school years, and will therefore dominate sports. If there is a bias in the spread of birth dates, we can expect a higher chance of shared birthdays. And often there is a bias. For example, a sizeable proportion of babies are now born by caesarian section or induced. This tends to happen on weekdays (as maternity staff prefer not to work weekends), with the result that births are not spread as randomly throughout the year. If you take a section of 23 people born in the same 12-month period