The Birthday Paradox Explained |Why 23 People Are Enough |Probability & Statistics |Tanmay Kulkarni

[0:00]23 people in a room bet me two of them share a birthday. Most people take the other side. They lose that bet about half the time and I want to show you exactly how I know. Hi, I'm Tanmay Kulkarni in for Humanitarians AI. Three things could change that odds figure. How many people are in the group, how uneven the birthdays are, and how many days they have to land on. I'll change each one and watch the number. One of them barely matters, one matters the way you'd expect, and one decides almost everything. Hold a guess about which. The trick is to stop counting people and

[0:31]Count pairs. 23 people make 253 pairs. Each pair is a small chance and 253 small chances pile up faster than your gut expects. First change the people. I run 400,000 random groups of 23 and count how often a pair shares a day. 50.7% The exact calculation agrees. Now add people. With 10 about 12%, with 30 71%, with 50 97%. By 57 people past 99%. More people, more odds, just as you'd expect. One thing the claim does not say. It says some pair, not a pair with you. For somebody to share your birthday, you need 253 other people to reach even odds. With 22 others in the room, your

[1:12]Chance is under 6%. Same crowd, a different question. Second change how even the days are. So far every day was equally likely and that's false. I used federal counts of babies born each day. They come from the National Center for Health Statistics and the Social Security Administration and cover 2000 to 2014. The busiest day runs about 12% above average. February 29th runs at about a quarter of a normal day because it only exists one year in four. So I ran it again on the real calendar. 50.8% against 50.7 for even days. Uneven days can only raise the odds and they did by a tenth of a point at most.

[1:50]That's tiny. About 96% of days sit within 10% of average, so the real calendar is close enough to even that the model barely notices. The wrong assumption was forgiving. Keep that in mind because the third change breaks the model somewhere else entirely. Third change, how many days there are to land on, squeeze it. Not 23 people from anywhere in the year, but 23 people all born inside the same 90 days. 95.4% same group size, a quarter of the calendar, and the odds go from a coin flip to nearly certain. Let me slide the window. A full year 50.8, half a year 77, 90 days 95, 60

[2:29]Days 99. Born in the same month, better than 99.9%, and with a window of 7 days, 23 people can't avoid sharing. There aren't enough days. Flip it around and ask how many people you need for even odds. Across a whole year, 23. 90-day window, 12. 60 days, 10. A 30-day window, 7. A fortnight, 5. A week, 4. Here is what I almost missed. Does the uneven calendar still matter inside those windows? A tenth of a point at most, again. Seasons weren't the story. Of the three things, the head count moves the odds the way you'd expect. The evenness of the days barely moves them, and the number of days decides almost

[3:07]Everything. Look at those head counts again. 23 for 365 days, 12 for 90, 7 for 30. They follow a rule. Take the square root of the days, multiply by 1.2, add a half, and round. A year gives 23, 90 days gives 12, 30 days gives 7, a week gives 4. It is right for every window in this film, and never more than one person off anywhere from two slots to 2,000. A fair objection, classmates are all about the same age. But a school year with a cutoff date still contains every day of the calendar once, so it behaves like the full year, about 50.8%. The squeeze only happens if the group is

[3:42]Pinned to a narrow stretch of days. Now use it on something that isn't a birthday. Birth months, 12 slots. The square root of 12 is about 3 and 1/2 * 1.2 + 1/2 rounds to 5. So, five people are enough for even odds that two share a birth month if every month were equally likely. Four people give 43%, five give 62. Your turn. Everyone in a class picks a random number from 1 to 50. How many people before two probably match? Use the rule. Attempting wrong answer is half the slots, 25. Pause, work it out, and I'll say. The root of 50 is about 7 * 1.2 + 1/2

[4:27]Rounds to 9. Nine people give 53%, eight give 45, not 25. The slots set the scale and their square root sets the crowd. Next time someone quotes odds that two things match, ask how many slots they could fall into. The square root of that number is about how many it takes. Tanmay Kulkarni, in for Humanitarian's AI.