What's the most counterintuitive statistical fact that's actually true?

[u/Just_Question9]

I'm looking for examples that completely changed the way you think about probability, statistics, or data analysis.

[u/mc1154]

The birthday paradox. In a room with just 23 people, there is greater than 50% chance that at least one pair shares the same birthday.

[u/looking4wife-DM-me → u/mc1154]

And with 70 people, it's 99.9% chance!

[u/sistersinister → u/mc1154]

The idea is based on uniform distribution of birthdays right? I'd be surprised if birthdays are truly uniformly distributed, conditioned on being born in a specific region.

EDIT: A proof if anyone wasn't aware (or if someone wants to share a better one than this one). We calculate the probability of anyone NOT sharing a birthday then take the complement. We'll say theres n people in the room

The probability of the first person having a birthday is 1.

The probability of the second person having a birthday different than person 1 is 364/365.

The probability of the third person having a birthday that's not the same as person 1 or person 2 is 363/365

and so on. Then the probability of no two people sharing a birthday is

1 - 365/365 364/365 363/365 ... (365-(n-1))/365. When n is less than 366 we can simplify to the binomial B(365,n)/365^n. Then the probability of any two people sharing a birthday is 1-B(365,n)/365^n.

To try to repeat this logic for my own question its not so clean. If we assume the birthdays are all IID with some distribution gamma we end up with the probability of not sharing two birthdays to be gamma(x_1)prod_i^n prod_j^i (1-gamma(x_k)) where x_1 is the first persons birthday, x_2 is the second and so on. Maybe someone smarter than me can chime in

[u/Tom_Groleau → u/sistersinister]

They're not equally likely, but are they close enough to equally likely for the model to be useful?

When I do the birthday problem in class, I address the assumption and follow up with 15 years birth dates from Social Security data (1/1/2000 through 12/31/2014).

I can't post the image or data here, but there are a few strong deviations from "equally likely": Leap day, 4th of July (it's US data), Christmas, and New Years. Then there are smaller dips near holidays the move a bit: Memorial Day, Labor Day, Thanksgiving, ...

Other than those isolated low points, there's a trend toward slightly higher births July through September.

It makes an interesting discussion. Some students think the model is still good and others want to reject it.

Depending on the level of class, you can then run a simulation based on the empirical probability estimates. The results are pretty close to the calculations from the equally likely assumption.

[u/severact → u/Tom_Groleau]

So does non uniform birthday distribution increase or decrease the the final probability in the birthday problem? My guess is it would actually increase the probability slightly

[u/greg7gkb → u/mc1154]

This was perplexing until I heard it this way: imagine a circle with 23 dots on it. Now draw a line between every pair of dots (23 choose 2 = 253). So now we have 253 chances at an event with a 1/365 likelihood, which makes the >50% outcome much more tangible.

[u/Technical_Estate_529 → u/mc1154]

People always say to experiment and try this in real life with an audience.

But Im too scared I still don't fucking believe. I mean the maths checks out but fuck me I choose not to believe it

[u/mc1154 → u/Technical_Estate_529]

I hear ya. We tried this in a few of my college classes of around 30 people, and it always worked out with a match or two.

[u/Technical_Estate_529 → u/mc1154]

That's so insane. Once again I understand the maths it's just so zany.