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Imagine a group of people.

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How big do you think the group
would have to be

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before there's more than a 50% chance
that two people in the group

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have the same birthday?

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Assume for the sake of argument
that there are no twins,

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that every birthday is equally likely,

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and ignore leap years.

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Take a moment to think about it.

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The answer may seem surprisingly low.

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In a group of 23 people,

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there's a 50.73% chance that 
two people will share the same birthday.

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00:00:44,669 --> 00:00:47,239
But with 365 days in a year,

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how's it possible that you need such
a small group

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to get even odds of a shared birthday?

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Why is our intuition so wrong?

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To figure out the answer,

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let's look at one way a mathematician

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might calculate 
the odds of a birthday match.

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We can use a field of mathematics
known as combinatorics,

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which deals with the likelihoods
of different combinations.

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The first step is to flip the problem.

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Trying to calculate the odds 
of a match directly is challenging

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because there are many ways you
could get a birthday match in a group.

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Instead, it's easier to calculate the odds
that everyone's birthday is different.

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How does that help?

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Either there's a birthday match
in the group, or there isn't,

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so the odds of a match
and the odds of no match

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must add up to 100%.

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That means we can find 
the probability of a match

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by subtracting the probability
of no match from 100.

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To calculate the odds of no match,
start small.

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Calculate the odds that just one pair
of people have different birthdays.

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One day of the year will be
Person A's birthday,

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which leaves only 364 possible birthdays
for Person B.

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The probability of different birthdays
for A and B, or any pair of people,

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is 364 out of 365,

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about 0.997, or 99.7%, pretty high.

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Bring in Person C.

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The probability that she has 
a unique birthday in this small group

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is 363 out of 365

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because there are two birthdates
already accounted for by A and B.

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D's odds will be 362 out of 365,
and so on,

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all the way down to W's odds
of 343 out of 365.

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Multiply all of those terms together,

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and you'll get the probability
that no one shares a birthday.

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This works out to 0.4927,

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so there's a 49.27% chance that no one in
the group of 23 people shares a birthday.

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When we subtract that from 100,
we get a 50.73% chance

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of at least one birthday match,

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better than even odds.

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The key to such a high probability 
of a match in a relatively small group

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is the surprisingly large number
of possible pairs.

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As a group grows, the number of possible 
combinations gets bigger much faster.

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A group of five people 
has ten possible pairs.

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Each of the five people can be paired
with any of the other four.

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Half of those combinations are redundant

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because pairing Person A with Person B
is the same as pairing B with A,

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so we divide by two.

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By the same reasoning,

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a group of ten people has 45 pairs,

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and a group of 23 has 253.

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The number of pairs grows quadratically,

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meaning it's proportional to the square
of the number of people in the group.

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Unfortunately, our brains 
are notoriously bad

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at intuitively grasping 
non-linear functions.

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So it seems improbable at first that 23
people could produce 253 possible pairs.

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Once our brains accept that,
the birthday problem makes more sense.

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Every one of those 253 pairs is a chance
for a birthday match.

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For the same reason,
in a group of 70 people,

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there are 2,415 possible pairs,

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and the probability that two people
have the same birthday is more than 99.9%.

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The birthday problem is just one example
where math can show

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that things that seem impossible,

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like the same person winning
the lottery twice,

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actually aren't unlikely at all.

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Sometimes coincidences aren't
as coincidental as they seem.