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.
