Check your intuition: The birthday problem - David Knuffke
-
0:10 - 0:12Imagine a group of people.
-
0:12 - 0:14How big do you think the group
would have to be -
0:14 - 0:19before there's more than a 50% chance
that two people in the group -
0:19 - 0:21have the same birthday?
-
0:21 - 0:24Assume for the sake of argument
that there are no twins, -
0:24 - 0:27that every birthday is equally likely,
-
0:27 - 0:30and ignore leap years.
-
0:30 - 0:33Take a moment to think about it.
-
0:33 - 0:36The answer may seem surprisingly low.
-
0:36 - 0:38In a group of 23 people,
-
0:38 - 0:45there's a 50.73% chance that
two people will share the same birthday. -
0:45 - 0:47But with 365 days in a year,
-
0:47 - 0:50how's it possible that you need such
a small group -
0:50 - 0:54to get even odds of a shared birthday?
-
0:54 - 0:58Why is our intuition so wrong?
-
0:58 - 0:59To figure out the answer,
-
0:59 - 1:01let's look at one way a mathematician
-
1:01 - 1:05might calculate
the odds of a birthday match. -
1:05 - 1:09We can use a field of mathematics
known as combinatorics, -
1:09 - 1:14which deals with the likelihoods
of different combinations. -
1:14 - 1:17The first step is to flip the problem.
-
1:17 - 1:21Trying to calculate the odds
of a match directly is challenging -
1:21 - 1:25because there are many ways you
could get a birthday match in a group. -
1:25 - 1:31Instead, it's easier to calculate the odds
that everyone's birthday is different. -
1:31 - 1:33How does that help?
-
1:33 - 1:36Either there's a birthday match
in the group, or there isn't, -
1:36 - 1:38so the odds of a match
and the odds of no match -
1:38 - 1:42must add up to 100%.
-
1:42 - 1:44That means we can find
the probability of a match -
1:44 - 1:50by subtracting the probability
of no match from 100. -
1:50 - 1:54To calculate the odds of no match,
start small. -
1:54 - 1:58Calculate the odds that just one pair
of people have different birthdays. -
1:58 - 2:01One day of the year will be
Person A's birthday, -
2:01 - 2:06which leaves only 364 possible birthdays
for Person B. -
2:06 - 2:11The probability of different birthdays
for A and B, or any pair of people, -
2:11 - 2:14is 364 out of 365,
-
2:14 - 2:21about 0.997, or 99.7%, pretty high.
-
2:21 - 2:23Bring in Person C.
-
2:23 - 2:26The probability that she has
a unique birthday in this small group -
2:26 - 2:30is 363 out of 365
-
2:30 - 2:34because there are two birthdates
already accounted for by A and B. -
2:34 - 2:39D's odds will be 362 out of 365,
and so on, -
2:39 - 2:44all the way down to W's odds
of 343 out of 365. -
2:44 - 2:46Multiply all of those terms together,
-
2:46 - 2:51and you'll get the probability
that no one shares a birthday. -
2:51 - 2:54This works out to 0.4927,
-
2:54 - 3:01so there's a 49.27% chance that no one in
the group of 23 people shares a birthday. -
3:01 - 3:06When we subtract that from 100,
we get a 50.73% chance -
3:06 - 3:09of at least one birthday match,
-
3:09 - 3:12better than even odds.
-
3:12 - 3:16The key to such a high probability
of a match in a relatively small group -
3:16 - 3:20is the surprisingly large number
of possible pairs. -
3:20 - 3:26As a group grows, the number of possible
combinations gets bigger much faster. -
3:26 - 3:29A group of five people
has ten possible pairs. -
3:29 - 3:33Each of the five people can be paired
with any of the other four. -
3:33 - 3:35Half of those combinations are redundant
-
3:35 - 3:40because pairing Person A with Person B
is the same as pairing B with A, -
3:40 - 3:42so we divide by two.
-
3:42 - 3:43By the same reasoning,
-
3:43 - 3:46a group of ten people has 45 pairs,
-
3:46 - 3:50and a group of 23 has 253.
-
3:50 - 3:53The number of pairs grows quadratically,
-
3:53 - 3:58meaning it's proportional to the square
of the number of people in the group. -
3:58 - 4:01Unfortunately, our brains
are notoriously bad -
4:01 - 4:04at intuitively grasping
non-linear functions. -
4:04 - 4:11So it seems improbable at first that 23
people could produce 253 possible pairs. -
4:11 - 4:15Once our brains accept that,
the birthday problem makes more sense. -
4:15 - 4:20Every one of those 253 pairs is a chance
for a birthday match. -
4:20 - 4:23For the same reason,
in a group of 70 people, -
4:23 - 4:27there are 2,415 possible pairs,
-
4:27 - 4:33and the probability that two people
have the same birthday is more than 99.9%. -
4:33 - 4:37The birthday problem is just one example
where math can show -
4:37 - 4:39that things that seem impossible,
-
4:39 - 4:41like the same person winning
the lottery twice, -
4:41 - 4:45actually aren't unlikely at all.
-
4:45 - 4:49Sometimes coincidences aren't
as coincidental as they seem.
- Title:
- Check your intuition: The birthday problem - David Knuffke
- Description:
-
View full lesson: http://ed.ted.com/lessons/check-your-intuition-the-birthday-problem-david-knuffke
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 the group have the same birthday? The answer is … probably lower than you think. David Knuffke explains how the birthday problem exposes our often-poor intuition when it comes to probability.
Lesson by David Knuffke, animation by TED-Ed.
- Video Language:
- English
- Team:
- closed TED
- Project:
- TED-Ed
- Duration:
- 05:07
Krystian Aparta edited English subtitles for Check your intuition: The birthday problem - David Knuffke |