[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:10.05,0:00:11.93,Default,,0000,0000,0000,,Imagine a group of people.
Dialogue: 0,0:00:11.93,0:00:14.30,Default,,0000,0000,0000,,How big do you think the group\Nwould have to be
Dialogue: 0,0:00:14.30,0:00:18.78,Default,,0000,0000,0000,,before there's more than a 50% chance\Nthat two people in the group
Dialogue: 0,0:00:18.78,0:00:21.22,Default,,0000,0000,0000,,have the same birthday?
Dialogue: 0,0:00:21.22,0:00:24.19,Default,,0000,0000,0000,,Assume for the sake of argument\Nthat there are no twins,
Dialogue: 0,0:00:24.19,0:00:26.75,Default,,0000,0000,0000,,that every birthday is equally likely,
Dialogue: 0,0:00:26.75,0:00:29.98,Default,,0000,0000,0000,,and ignore leap years.
Dialogue: 0,0:00:29.98,0:00:33.05,Default,,0000,0000,0000,,Take a moment to think about it.
Dialogue: 0,0:00:33.05,0:00:35.91,Default,,0000,0000,0000,,The answer may seem surprisingly low.
Dialogue: 0,0:00:35.91,0:00:37.71,Default,,0000,0000,0000,,In a group of 23 people,
Dialogue: 0,0:00:37.71,0:00:44.67,Default,,0000,0000,0000,,there's a 50.73% chance that \Ntwo people will share the same birthday.
Dialogue: 0,0:00:44.67,0:00:47.24,Default,,0000,0000,0000,,But with 365 days in a year,
Dialogue: 0,0:00:47.24,0:00:50.49,Default,,0000,0000,0000,,how's it possible that you need such\Na small group
Dialogue: 0,0:00:50.49,0:00:53.70,Default,,0000,0000,0000,,to get even odds of a shared birthday?
Dialogue: 0,0:00:53.70,0:00:58.16,Default,,0000,0000,0000,,Why is our intuition so wrong?
Dialogue: 0,0:00:58.16,0:00:59.50,Default,,0000,0000,0000,,To figure out the answer,
Dialogue: 0,0:00:59.50,0:01:01.39,Default,,0000,0000,0000,,let's look at one way a mathematician
Dialogue: 0,0:01:01.39,0:01:05.22,Default,,0000,0000,0000,,might calculate \Nthe odds of a birthday match.
Dialogue: 0,0:01:05.22,0:01:09.11,Default,,0000,0000,0000,,We can use a field of mathematics\Nknown as combinatorics,
Dialogue: 0,0:01:09.11,0:01:14.42,Default,,0000,0000,0000,,which deals with the likelihoods\Nof different combinations.
Dialogue: 0,0:01:14.42,0:01:16.95,Default,,0000,0000,0000,,The first step is to flip the problem.
Dialogue: 0,0:01:16.95,0:01:21.33,Default,,0000,0000,0000,,Trying to calculate the odds \Nof a match directly is challenging
Dialogue: 0,0:01:21.33,0:01:25.23,Default,,0000,0000,0000,,because there are many ways you\Ncould get a birthday match in a group.
Dialogue: 0,0:01:25.23,0:01:31.39,Default,,0000,0000,0000,,Instead, it's easier to calculate the odds\Nthat everyone's birthday is different.
Dialogue: 0,0:01:31.39,0:01:32.82,Default,,0000,0000,0000,,How does that help?
Dialogue: 0,0:01:32.82,0:01:35.74,Default,,0000,0000,0000,,Either there's a birthday match\Nin the group, or there isn't,
Dialogue: 0,0:01:35.74,0:01:38.46,Default,,0000,0000,0000,,so the odds of a match\Nand the odds of no match
Dialogue: 0,0:01:38.46,0:01:41.86,Default,,0000,0000,0000,,must add up to 100%.
Dialogue: 0,0:01:41.86,0:01:44.27,Default,,0000,0000,0000,,That means we can find \Nthe probability of a match
Dialogue: 0,0:01:44.27,0:01:50.38,Default,,0000,0000,0000,,by subtracting the probability\Nof no match from 100.
Dialogue: 0,0:01:50.38,0:01:53.81,Default,,0000,0000,0000,,To calculate the odds of no match,\Nstart small.
Dialogue: 0,0:01:53.81,0:01:58.28,Default,,0000,0000,0000,,Calculate the odds that just one pair\Nof people have different birthdays.
Dialogue: 0,0:01:58.28,0:02:00.63,Default,,0000,0000,0000,,One day of the year will be\NPerson A's birthday,
Dialogue: 0,0:02:00.63,0:02:06.02,Default,,0000,0000,0000,,which leaves only 364 possible birthdays\Nfor Person B.
Dialogue: 0,0:02:06.02,0:02:10.59,Default,,0000,0000,0000,,The probability of different birthdays\Nfor A and B, or any pair of people,
Dialogue: 0,0:02:10.59,0:02:14.41,Default,,0000,0000,0000,,is 364 out of 365,
Dialogue: 0,0:02:14.41,0:02:20.51,Default,,0000,0000,0000,,about 0.997, or 99.7%, pretty high.
Dialogue: 0,0:02:20.51,0:02:22.56,Default,,0000,0000,0000,,Bring in Person C.
Dialogue: 0,0:02:22.56,0:02:25.79,Default,,0000,0000,0000,,The probability that she has \Na unique birthday in this small group
Dialogue: 0,0:02:25.79,0:02:29.53,Default,,0000,0000,0000,,is 363 out of 365
Dialogue: 0,0:02:29.53,0:02:33.96,Default,,0000,0000,0000,,because there are two birthdates\Nalready accounted for by A and B.
Dialogue: 0,0:02:33.96,0:02:38.58,Default,,0000,0000,0000,,D's odds will be 362 out of 365,\Nand so on,
Dialogue: 0,0:02:38.58,0:02:44.47,Default,,0000,0000,0000,,all the way down to W's odds\Nof 343 out of 365.
Dialogue: 0,0:02:44.47,0:02:46.38,Default,,0000,0000,0000,,Multiply all of those terms together,
Dialogue: 0,0:02:46.38,0:02:50.94,Default,,0000,0000,0000,,and you'll get the probability\Nthat no one shares a birthday.
Dialogue: 0,0:02:50.94,0:02:54.06,Default,,0000,0000,0000,,This works out to 0.4927,
Dialogue: 0,0:02:54.06,0:03:01.36,Default,,0000,0000,0000,,so there's a 49.27% chance that no one in\Nthe group of 23 people shares a birthday.
Dialogue: 0,0:03:01.36,0:03:05.96,Default,,0000,0000,0000,,When we subtract that from 100,\Nwe get a 50.73% chance
Dialogue: 0,0:03:05.96,0:03:08.70,Default,,0000,0000,0000,,of at least one birthday match,
Dialogue: 0,0:03:08.70,0:03:11.96,Default,,0000,0000,0000,,better than even odds.
Dialogue: 0,0:03:11.96,0:03:16.14,Default,,0000,0000,0000,,The key to such a high probability \Nof a match in a relatively small group
Dialogue: 0,0:03:16.14,0:03:20.32,Default,,0000,0000,0000,,is the surprisingly large number\Nof possible pairs.
Dialogue: 0,0:03:20.32,0:03:26.02,Default,,0000,0000,0000,,As a group grows, the number of possible \Ncombinations gets bigger much faster.
Dialogue: 0,0:03:26.02,0:03:29.20,Default,,0000,0000,0000,,A group of five people \Nhas ten possible pairs.
Dialogue: 0,0:03:29.20,0:03:32.90,Default,,0000,0000,0000,,Each of the five people can be paired\Nwith any of the other four.
Dialogue: 0,0:03:32.90,0:03:34.84,Default,,0000,0000,0000,,Half of those combinations are redundant
Dialogue: 0,0:03:34.84,0:03:39.62,Default,,0000,0000,0000,,because pairing Person A with Person B\Nis the same as pairing B with A,
Dialogue: 0,0:03:39.62,0:03:41.68,Default,,0000,0000,0000,,so we divide by two.
Dialogue: 0,0:03:41.68,0:03:43.04,Default,,0000,0000,0000,,By the same reasoning,
Dialogue: 0,0:03:43.04,0:03:45.84,Default,,0000,0000,0000,,a group of ten people has 45 pairs,
Dialogue: 0,0:03:45.84,0:03:49.84,Default,,0000,0000,0000,,and a group of 23 has 253.
Dialogue: 0,0:03:49.84,0:03:52.90,Default,,0000,0000,0000,,The number of pairs grows quadratically,
Dialogue: 0,0:03:52.90,0:03:57.66,Default,,0000,0000,0000,,meaning it's proportional to the square\Nof the number of people in the group.
Dialogue: 0,0:03:57.66,0:04:00.97,Default,,0000,0000,0000,,Unfortunately, our brains \Nare notoriously bad
Dialogue: 0,0:04:00.97,0:04:04.45,Default,,0000,0000,0000,,at intuitively grasping \Nnon-linear functions.
Dialogue: 0,0:04:04.45,0:04:11.24,Default,,0000,0000,0000,,So it seems improbable at first that 23\Npeople could produce 253 possible pairs.
Dialogue: 0,0:04:11.24,0:04:15.27,Default,,0000,0000,0000,,Once our brains accept that,\Nthe birthday problem makes more sense.
Dialogue: 0,0:04:15.27,0:04:20.14,Default,,0000,0000,0000,,Every one of those 253 pairs is a chance\Nfor a birthday match.
Dialogue: 0,0:04:20.14,0:04:22.90,Default,,0000,0000,0000,,For the same reason,\Nin a group of 70 people,
Dialogue: 0,0:04:22.90,0:04:26.62,Default,,0000,0000,0000,,there are 2,415 possible pairs,
Dialogue: 0,0:04:26.62,0:04:33.34,Default,,0000,0000,0000,,and the probability that two people\Nhave the same birthday is more than 99.9%.
Dialogue: 0,0:04:33.34,0:04:36.71,Default,,0000,0000,0000,,The birthday problem is just one example\Nwhere math can show
Dialogue: 0,0:04:36.71,0:04:38.92,Default,,0000,0000,0000,,that things that seem impossible,
Dialogue: 0,0:04:38.92,0:04:41.41,Default,,0000,0000,0000,,like the same person winning\Nthe lottery twice,
Dialogue: 0,0:04:41.41,0:04:44.55,Default,,0000,0000,0000,,actually aren't unlikely at all.
Dialogue: 0,0:04:44.55,0:04:48.87,Default,,0000,0000,0000,,Sometimes coincidences aren't\Nas coincidental as they seem.