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  "events": [ {
    "tStartMs": 10048,
    "dDurationMs": 1885,
    "segs": [ {
      "utf8": "Imagine a group of people."
    } ]
  }, {
    "tStartMs": 11933,
    "dDurationMs": 2371,
    "segs": [ {
      "utf8": "How big do you think the group\nwould have to be"
    } ]
  }, {
    "tStartMs": 14304,
    "dDurationMs": 4474,
    "segs": [ {
      "utf8": "before there's more than a 50% chance\nthat two people in the group"
    } ]
  }, {
    "tStartMs": 18778,
    "dDurationMs": 2440,
    "segs": [ {
      "utf8": "have the same birthday?"
    } ]
  }, {
    "tStartMs": 21218,
    "dDurationMs": 2969,
    "segs": [ {
      "utf8": "Assume for the sake of argument\nthat there are no twins,"
    } ]
  }, {
    "tStartMs": 24187,
    "dDurationMs": 2561,
    "segs": [ {
      "utf8": "that every birthday is equally likely,"
    } ]
  }, {
    "tStartMs": 26748,
    "dDurationMs": 3229,
    "segs": [ {
      "utf8": "and ignore leap years."
    } ]
  }, {
    "tStartMs": 29977,
    "dDurationMs": 3072,
    "segs": [ {
      "utf8": "Take a moment to think about it."
    } ]
  }, {
    "tStartMs": 33049,
    "dDurationMs": 2859,
    "segs": [ {
      "utf8": "The answer may seem surprisingly low."
    } ]
  }, {
    "tStartMs": 35908,
    "dDurationMs": 1800,
    "segs": [ {
      "utf8": "In a group of 23 people,"
    } ]
  }, {
    "tStartMs": 37708,
    "dDurationMs": 6961,
    "segs": [ {
      "utf8": "there's a 50.73% chance that \ntwo people will share the same birthday."
    } ]
  }, {
    "tStartMs": 44669,
    "dDurationMs": 2570,
    "segs": [ {
      "utf8": "But with 365 days in a year,"
    } ]
  }, {
    "tStartMs": 47239,
    "dDurationMs": 3250,
    "segs": [ {
      "utf8": "how's it possible that you need such\na small group"
    } ]
  }, {
    "tStartMs": 50489,
    "dDurationMs": 3211,
    "segs": [ {
      "utf8": "to get even odds of a shared birthday?"
    } ]
  }, {
    "tStartMs": 53700,
    "dDurationMs": 4456,
    "segs": [ {
      "utf8": "Why is our intuition so wrong?"
    } ]
  }, {
    "tStartMs": 58156,
    "dDurationMs": 1342,
    "segs": [ {
      "utf8": "To figure out the answer,"
    } ]
  }, {
    "tStartMs": 59498,
    "dDurationMs": 1891,
    "segs": [ {
      "utf8": "let's look at one way a mathematician"
    } ]
  }, {
    "tStartMs": 61389,
    "dDurationMs": 3829,
    "segs": [ {
      "utf8": "might calculate \nthe odds of a birthday match."
    } ]
  }, {
    "tStartMs": 65218,
    "dDurationMs": 3892,
    "segs": [ {
      "utf8": "We can use a field of mathematics\nknown as combinatorics,"
    } ]
  }, {
    "tStartMs": 69110,
    "dDurationMs": 5309,
    "segs": [ {
      "utf8": "which deals with the likelihoods\nof different combinations."
    } ]
  }, {
    "tStartMs": 74419,
    "dDurationMs": 2531,
    "segs": [ {
      "utf8": "The first step is to flip the problem."
    } ]
  }, {
    "tStartMs": 76950,
    "dDurationMs": 4380,
    "segs": [ {
      "utf8": "Trying to calculate the odds \nof a match directly is challenging"
    } ]
  }, {
    "tStartMs": 81330,
    "dDurationMs": 3899,
    "segs": [ {
      "utf8": "because there are many ways you\ncould get a birthday match in a group."
    } ]
  }, {
    "tStartMs": 85229,
    "dDurationMs": 6160,
    "segs": [ {
      "utf8": "Instead, it's easier to calculate the odds\nthat everyone's birthday is different."
    } ]
  }, {
    "tStartMs": 91389,
    "dDurationMs": 1431,
    "segs": [ {
      "utf8": "How does that help?"
    } ]
  }, {
    "tStartMs": 92820,
    "dDurationMs": 2921,
    "segs": [ {
      "utf8": "Either there's a birthday match\nin the group, or there isn't,"
    } ]
  }, {
    "tStartMs": 95741,
    "dDurationMs": 2720,
    "segs": [ {
      "utf8": "so the odds of a match\nand the odds of no match"
    } ]
  }, {
    "tStartMs": 98461,
    "dDurationMs": 3399,
    "segs": [ {
      "utf8": "must add up to 100%."
    } ]
  }, {
    "tStartMs": 101860,
    "dDurationMs": 2411,
    "segs": [ {
      "utf8": "That means we can find \nthe probability of a match"
    } ]
  }, {
    "tStartMs": 104271,
    "dDurationMs": 6110,
    "segs": [ {
      "utf8": "by subtracting the probability\nof no match from 100."
    } ]
  }, {
    "tStartMs": 110381,
    "dDurationMs": 3425,
    "segs": [ {
      "utf8": "To calculate the odds of no match,\nstart small."
    } ]
  }, {
    "tStartMs": 113806,
    "dDurationMs": 4475,
    "segs": [ {
      "utf8": "Calculate the odds that just one pair\nof people have different birthdays."
    } ]
  }, {
    "tStartMs": 118281,
    "dDurationMs": 2351,
    "segs": [ {
      "utf8": "One day of the year will be\nPerson A's birthday,"
    } ]
  }, {
    "tStartMs": 120632,
    "dDurationMs": 5390,
    "segs": [ {
      "utf8": "which leaves only 364 possible birthdays\nfor Person B."
    } ]
  }, {
    "tStartMs": 126022,
    "dDurationMs": 4570,
    "segs": [ {
      "utf8": "The probability of different birthdays\nfor A and B, or any pair of people,"
    } ]
  }, {
    "tStartMs": 130592,
    "dDurationMs": 3820,
    "segs": [ {
      "utf8": "is 364 out of 365,"
    } ]
  }, {
    "tStartMs": 134412,
    "dDurationMs": 6102,
    "segs": [ {
      "utf8": "about 0.997, or 99.7%, pretty high."
    } ]
  }, {
    "tStartMs": 140514,
    "dDurationMs": 2048,
    "segs": [ {
      "utf8": "Bring in Person C."
    } ]
  }, {
    "tStartMs": 142562,
    "dDurationMs": 3231,
    "segs": [ {
      "utf8": "The probability that she has \na unique birthday in this small group"
    } ]
  }, {
    "tStartMs": 145793,
    "dDurationMs": 3739,
    "segs": [ {
      "utf8": "is 363 out of 365"
    } ]
  }, {
    "tStartMs": 149532,
    "dDurationMs": 4432,
    "segs": [ {
      "utf8": "because there are two birthdates\nalready accounted for by A and B."
    } ]
  }, {
    "tStartMs": 153964,
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    "segs": [ {
      "utf8": "D's odds will be 362 out of 365,\nand so on,"
    } ]
  }, {
    "tStartMs": 158582,
    "dDurationMs": 5892,
    "segs": [ {
      "utf8": "all the way down to W's odds\nof 343 out of 365."
    } ]
  }, {
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    "dDurationMs": 1911,
    "segs": [ {
      "utf8": "Multiply all of those terms together,"
    } ]
  }, {
    "tStartMs": 166385,
    "dDurationMs": 4557,
    "segs": [ {
      "utf8": "and you'll get the probability\nthat no one shares a birthday."
    } ]
  }, {
    "tStartMs": 170942,
    "dDurationMs": 3122,
    "segs": [ {
      "utf8": "This works out to 0.4927,"
    } ]
  }, {
    "tStartMs": 174064,
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    "segs": [ {
      "utf8": "so there's a 49.27% chance that no one in\nthe group of 23 people shares a birthday."
    } ]
  }, {
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    "segs": [ {
      "utf8": "When we subtract that from 100,\nwe get a 50.73% chance"
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  }, {
    "tStartMs": 185955,
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    "segs": [ {
      "utf8": "of at least one birthday match,"
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  }, {
    "tStartMs": 188701,
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    "segs": [ {
      "utf8": "better than even odds."
    } ]
  }, {
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    "segs": [ {
      "utf8": "The key to such a high probability \nof a match in a relatively small group"
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  }, {
    "tStartMs": 196144,
    "dDurationMs": 4181,
    "segs": [ {
      "utf8": "is the surprisingly large number\nof possible pairs."
    } ]
  }, {
    "tStartMs": 200325,
    "dDurationMs": 5692,
    "segs": [ {
      "utf8": "As a group grows, the number of possible \ncombinations gets bigger much faster."
    } ]
  }, {
    "tStartMs": 206017,
    "dDurationMs": 3179,
    "segs": [ {
      "utf8": "A group of five people \nhas ten possible pairs."
    } ]
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    "dDurationMs": 3709,
    "segs": [ {
      "utf8": "Each of the five people can be paired\nwith any of the other four."
    } ]
  }, {
    "tStartMs": 212905,
    "dDurationMs": 1930,
    "segs": [ {
      "utf8": "Half of those combinations are redundant"
    } ]
  }, {
    "tStartMs": 214835,
    "dDurationMs": 4780,
    "segs": [ {
      "utf8": "because pairing Person A with Person B\nis the same as pairing B with A,"
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  }, {
    "tStartMs": 219615,
    "dDurationMs": 2070,
    "segs": [ {
      "utf8": "so we divide by two."
    } ]
  }, {
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    "segs": [ {
      "utf8": "By the same reasoning,"
    } ]
  }, {
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    "segs": [ {
      "utf8": "a group of ten people has 45 pairs,"
    } ]
  }, {
    "tStartMs": 225836,
    "dDurationMs": 3999,
    "segs": [ {
      "utf8": "and a group of 23 has 253."
    } ]
  }, {
    "tStartMs": 229835,
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    "segs": [ {
      "utf8": "The number of pairs grows quadratically,"
    } ]
  }, {
    "tStartMs": 232905,
    "dDurationMs": 4760,
    "segs": [ {
      "utf8": "meaning it's proportional to the square\nof the number of people in the group."
    } ]
  }, {
    "tStartMs": 237665,
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    "segs": [ {
      "utf8": "Unfortunately, our brains \nare notoriously bad"
    } ]
  }, {
    "tStartMs": 240966,
    "dDurationMs": 3481,
    "segs": [ {
      "utf8": "at intuitively grasping \nnon-linear functions."
    } ]
  }, {
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    "segs": [ {
      "utf8": "So it seems improbable at first that 23\npeople could produce 253 possible pairs."
    } ]
  }, {
    "tStartMs": 251235,
    "dDurationMs": 4032,
    "segs": [ {
      "utf8": "Once our brains accept that,\nthe birthday problem makes more sense."
    } ]
  }, {
    "tStartMs": 255267,
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    "segs": [ {
      "utf8": "Every one of those 253 pairs is a chance\nfor a birthday match."
    } ]
  }, {
    "tStartMs": 260135,
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    "segs": [ {
      "utf8": "For the same reason,\nin a group of 70 people,"
    } ]
  }, {
    "tStartMs": 262897,
    "dDurationMs": 3719,
    "segs": [ {
      "utf8": "there are 2,415 possible pairs,"
    } ]
  }, {
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    "segs": [ {
      "utf8": "and the probability that two people\nhave the same birthday is more than 99.9%."
    } ]
  }, {
    "tStartMs": 273337,
    "dDurationMs": 3370,
    "segs": [ {
      "utf8": "The birthday problem is just one example\nwhere math can show"
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  }, {
    "tStartMs": 276707,
    "dDurationMs": 2210,
    "segs": [ {
      "utf8": "that things that seem impossible,"
    } ]
  }, {
    "tStartMs": 278917,
    "dDurationMs": 2493,
    "segs": [ {
      "utf8": "like the same person winning\nthe lottery twice,"
    } ]
  }, {
    "tStartMs": 281410,
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    "segs": [ {
      "utf8": "actually aren't unlikely at all."
    } ]
  }, {
    "tStartMs": 284551,
    "dDurationMs": 4317,
    "segs": [ {
      "utf8": "Sometimes coincidences aren't\nas coincidental as they seem."
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