from -
MARTIN GARDNER
Three hundred years ago, a gambler asked Blaise
Pascal, the French mathematician, how to calculate the odds on certain dice
throws. Pascal’s answers were the beginning of “probability theory,” now one of
the fastest-growing branches of mathematics. Physicists use it to compute the probable path of a
neutron through heavy water, geneticists to determine the likelihood that a
couple will have blue-eyed children. Businessmen, economists, politicians,
military leaders - it is hard to
think of a profession in which probability theory is not applicable.
In daily life we make countless decisions based
on intuitive, common-sense estimates of probability; most of the time these
estimates are fairly reliable. Yet experts have found all sorts of curious
situations in which the actual probabilities differ startlingly from what we
expect.
One of the most difficult of all probability
calculations to believe is what mathematicians call the birthday paradox. Say
that you are attending a party at which 23
people are present. What is the probability that two of you were born on
the same day of the same month? Intuitively, you feel that the probability is
low. Actually, the chances are about even.
One way to work this out: Take any two people, and the chances are 364 out of 365
- or, there is a 364/365 probability - that their birth dates won’t
match. The chance that a third person’s birth date will miss both of theirs is 363/365; a
fourth person’s 362/365; and so on. Then, since the probability that all of
a series of separate events will occur (meaning, in this case, that all the
birthdays will be different) is the product of the probabilities for each
individual event, you multiply all these fractions together. As it turns out,
by the time you have 23 people
in the series, the probability fills below half; in other words, it becomes a
trifle better than even chance that, out of 23 birth dates, two will coincide.
With more people, the probability of a match goes up rapidly. For 30 people, the probability is better
than 7/10, or odds of 7 to 3 in favour of two coinciding dates. With 50 people,
the chances are better than 97 out of 100!
This seems so contrary to common sense that you may want to test it when you
are at a gathering of 23 or more
people. Or go through a Who’s Who and check 30 names at random. Seven times out of ten you’ll find a pair of
matching birth dates.
Professional gamblers know all sorts of “sucker
bets” based on the same principle as the birthday paradox. A gambler will bet
you at even odds, for example, that of the number plates on the next 20 passing cars, at least two will
match each other in their last two numerals. It sounds like a good bet? The
odds actually are 7 to 1 in his
favour.To work out the probability of any one outcome,
you need to know the number of different, equally possible outcomes. When you
toss a coin there are just two possibilities - heads or tails - so the
probability of flipping either is 1/2. In less simple situations,
however, it is easy to overlook
some of the possible outcomes and so arrive at a false probability estimate.
For example: What is the probability that, if a
family has three children, they will all be of the same sex? One might reason:
“At least two of the children are certain to be of the same sex. The third,
then, will either match them or it won’t
- so the probability that all three will be alike must be 1/2.” But
consider all the possible combinations, letting B stand for boy, G for girl:
BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. Only two of those eight sequences -
BBB and GGG - are all alike. The correct probability, therefore, that the three
children will be of the same sex is 2/8,
or 1/4.
Suppose a couple plans to have four offspring.
Which is more likely: three children of one sex and one of the other, or two
boys and two girls? Most people guess two and two. But when we list all the
possible combinations - there are 16 - we see that in six cases there is a
two-two split, but that there are eight sequences with a three-one split. So
the probability is 1
/2 that the sexes will be split three and one. Our
intuition has played us false again!
Another way of going wrong when estimating
probabilities is to assume that certain events are related when they are not.
Many people imagine that if a coin comes up heads several times in a row, the
odds will favour tails on the next toss. Not so. No matter how many times a
coin lands heads up, the probability of heads on the next toss remains 1/2.
Dozens of ridiculous systems for playing roulette and other games of chance are
based on this “gamblers’ fallacy” that previous results have an effect on
future results.
Then there was the man who thought he could
protect himself on plane trips by taking a harmless bomb along in his luggage
(very unlikely these days). He reasoned that the odds against one person taking
a bomb aboard were high, but that the odds against
two people doing it
were surely astronomical. His intuition was strong - but his grasp of
statistics and probabilities was, to say the least, flawed.
So the next
time you’re hedging your bets, be brave... ~ SB
