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Why is the probability of an 'impossible' event so high?


You will all have come across newspaper articles and television reports of events that are reported to be 'one in a million' , 'one in a billion' or maybe even 'one in a trillion'. But it is usually the case that the probability of such events is nowhere near as low as stated. In fact, it is usually the case that such events are so common that it would be more newsworthy if it did NOT happen.

As an example, I am sure you will have read a story like the following:

"Mother gives birth to 8th child, all of whom are boys - less than one in a billion probability!!!"

The fallacy here, as in all such stories, is to confuse the specific with the general. The probability of a specific mother (for example, YOUR mother) giving birth THIS year to her 8th child, all of which are boys is indeed very low (as I will explain below).  But the probability of this happening to at least one mother in the UK is almost a certainty.

Why?

In any given year there 
are approximately  700,000 births in the UK. Among these approximately 1,000 are to mothers having their 8th child.  Now, in a family of 8 children, the probability that all 8 are boys is 1 in 256 (the probability that the first is a boy is 1/2, the probability that the second is also a boy is 1/2 times 1/2, the probability that the third is also a boy is 1/2 times 1/2  times 1/2 etc).  On average, therefore, it is likely that in any given year there will be about FOUR mothers giving birth to their 8th child all of whom are boys. This is, therefore, hardly a newsworthy event. In fact it would be FAR more newsworthy if, in any given year, there was NO case of a mother giving birth to her 8th child all of which are boys.

That's because the probability that NONE of the 1000 families have all children  boys is about  0.02. To calculate this, we note that the probability that any one family are not all boys is 255/256 which is equal to 0.9961.  The probability that any two families are not all boys is that number times itself.
The probability that all 1000 families are not all boys is that number times itself 1000 times (i.e. 'to the power of 1000'). That works out at just below 0.02. So there is just a 2% chance that there will be no family all of whom are boys. This means there is a 98% chance that at least one family of 8 that will consist entirely of boys.

So where does the one in a million or one in billion come from? Well first let's just restrict ourselves to one of the specific 700,000 mothers giving birth this year. For any such mother chosen at random, the probability that she will be having her 8th child, is just 1 in 700. Now, in a family of 8 children, the probability that all 8 are boys is 1 in 256.  So the probability that the chosen mother gives birth to 8 boys is 1 divided by 700 divided by 256. This is about one in 180,000. But the probability of any specific mother in the UK (of whom there are about 15 million) giving birth at all THIS year is about 1 in 20, so you need to divide the one in 180,000 probability by 20 to arrive at a figure of about one in four million. If, in addition, you decide to 'narrow' the focus on to, say, the probability that any specific mother gives birth to her 8th boy in THIS particular week or even day then you can see how easy it is to argue that such a probability is less than one in a billion.  

There are MANY similar examples of supposedly 'impossible' events that the media loves to report on:

"Amercian woman wins lottery jackpot TWICE -  1 in 169000000000000 chance".  Again the chance of any specific person winning the jackpot a second time in a  specific week is indeed that low. But in, say, a 20 year period in a country like the US, the probability that at least one person wins the jackpot twice is actually again almost a certainty.

"Long-lost brothers die on the same day". In the UK alone there are probably 500,000 pairs of brothers who never see each other.  Any pair of brothers are likely to be close in age and so are likely to die within, say 10 years of each other -  about 4000 days. So the probability of any specific pair of brothers dying on the same day is  about 1 divided by 4000.  But the probability of at least one pair dying on the same day is one minus the probability that NONE of the 500,000 pairs of brothers dies on the same day. This latter probability is 3999/4000 to the power of 500,000.  This number is so small that it is more likely that Martians will be represented in the next Olympic games.

Family with three children born on the same day: Why this story should NOT have made the news

Another interesting cases involving the probability of seeing the same truck in two different cities.



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Norman Fenton


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