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imagine a group of people how big do you
think the group would have to be before
there's more than a 50% chance that two
people in group have the same birthday
assume for the sake of argument that
there are no twins that my birthday is
equally likely and ignore leap years
take a moment to think about it the
answer may seem surprisingly low in a
group of 23 people there is a 50 point 7
3 percent chance that two people will
share the same birthday
but with 365 days in a year how is it
possible that you need such a small
group to get even odds of a shared
birthday why is our intuition so wrong
to figure out the answer let's look at
one way a mathematician might sakya late
the odds of a birthday match we can use
a field of mathematics known as
combinatorics which deals with the
likelihoods of different combinations
the first step is to flip the problem
trying to calculate the odds of a match
directly is challenging because there
are many ways you could get a birthday
match in a group instead it's easier to
calculate the odds that everyone's
birthday is different how does that help
either there's a birthday match in the
group or there isn't so the odds of a
match and the odds of no match must add
up to 100% that means we can find the
probability of a match by subtracting
the probability of no match from 100 to
calculate the odds of no match start
small calculate the odds that just one
pair of people have different birthdays
one day in the year will be person as
birthday which leaves only 364 possible
birthdays for person B the probability
of different birthdays for a and B or
any pair of people is 364 out of 365
about 0.997 for 99.7% pretty hot bring
him for see see the probability that she
has a unique birthday in this small
group is 363 out of 365 because there
are two birth dates already accounted
for by a and D these odds will be 362
out of 365 and so on all the way down to
W side of 343 out of 365 multiply all of
those terms together and you'll get the
probability that no one shares a
birthday this works out to 0.49 to 7 so
there's a forty nine point two seven
percent chance that no one in the group
of 23 people shares a birthday when we
subtract that from 100 we get a 50 point
seven three percent chance of at least
wrong birthday match better than Egham
on the key to such a high probability of
a match in a relatively small group is
the surprisingly large number of
possible cans as a group grows the
number of possible combinations gets
bigger much faster a group of 5 people
has 10 possible pairs each of the 5
people can be paired with any of the
other 4 half of those combinations are
redundant because pairing person a with
person B is a same as pairing B with a
so we divide by 2 by the same reasoning
a group of 10 people has 45 pairs and a
group of 23 has 253 the number of pairs
grows quadratically meaning it's
proportional to the square of the number
of people in the group unfortunately our
brains are notoriously bad at
intuitively grasping nonlinear functions
so it seems improbable at first that 23
people could produce 253 possible pairs
once our brains accept that the birthday
problem makes more sense every one of
those 253 pairs is a chance for a
birthday match
for the same reason in a group of 70
people there are 2415 possible pairs and
the probability that two people have the
same birthday is more than 99.9 percent
the birthday problem is just one example
where math can show that things that
seem impossible like the same person
winning the lottery twice actually
aren't unlikely at all sometimes
coincidences aren't as coincidental as
they seem
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