Under what circumstances would one in four be the correct answer to the question of what is the probability of getting three heads in three tosses of a coin?
Well, if you had already completed one toss, which came down heads.
But if you were asked the question before the first toss had taken place, one in four would be the wrong answer. The probability of getting three heads in three tosses would be one in eight.
Something of this sort lies behind the accusation that a popular newspaper got the odds wrong when considering a ‘miracle coincidence’ of three babies born to the same family at exactly 7.43 on the clock.
The idea is that the question being addressed is What is the probability that three children would all be born at the same time, that is, any time will do, it doesn’t have to be 7.43, it could just as well be 9.58 or any of the other 718 minutes on the twelve hour clock.
In terms of tossing a coin, this is the equivalent of asking, before we began, for the probability of getting the same result in three tosses, indifferent to whether that was three heads or three tails. We specify heads only after the first toss comes down.
For the family, the odds started counting when the first baby was born.
But why not carry this argument further, move to after the second baby was born at 7.43. Then the chance of getting three in a row would drop to only one in 720, not a miracle story for the paper at all.
This very common confusion is what lies behind the belief that there is a Law of Averages.
Since it is the rule that two heads plus one tail is more likely than three heads in any three tosses of a coin, then two heads in a row must make it more likely that the next toss will produce a tail!
The need to change the calculus of probability, depending on where you stand in the process – how much you already know – also catches out some people who should know better. If your sample of ten produces a result which, though interesting is not significant, it is not correct to carry on sampling until, with the same formula, P attains the magic number, then stop, proclaiming Eureka. (There are of course methods for doing this correctly – much needed for example in stopping clinical trials which may be doing harm, or in minimising the sample size if testing involves destruction of the item being tested).
Suppose our father were a betting man – nothing heavy, just likes the odd flutter. When his wife first gave him the news that pregnancy was confirmed, he went to the bookies & asked for odds on his first child being born at 7.43 – perhaps that time or those numbers have some special significance for him. What odds would the betmaker offer? Would he place any conditions – bet voided if the birth is by Caesarean section, for example?
Is there indeed any strict medico-scientific (or, indeed, religious) defintion of the time of birth? When the head emerges (at least the worst is over)? At the moment when the midwife exclaims Congratulations! It’s a girl? When the cord is cut? When the baby is weighed? Or when the baby cries?
And how many attendants really record the time of birth (by the method nature intended) to the nearest minute? – I would expect a distinct bias towards numbers ending in 0 or 5.
But what if the proud father-to-be asked for odds on his first three children all being born at 7.43? Would the bookie offer odds, perhaps discounting the 300 million to one to allow for the probability that the happy couple may not in fact manage to have three children? Or would he smell a rat & decline to accept the bet?
The element of this story which makes me most inclined to detect a whiff in the air is in fact the very precision of the time – plus the fact that the father had the number tattooed on his arm.
