1/

Get a cup of coffee.

Let's talk about the Birthday Paradox.

This is a simple exercise in probability.

But from it, we can learn so much about life.

About strategic problem solving.

About non-linear thinking -- convexity, concavity, S curves, etc.

So let's dive in!

2/

Suppose we came across a "30 under 30" Forbes list.

The list features 30 highly accomplished people.

What are the chances that at least 2 of these 30 share the same birthday?

Same birthday means they were born on the same day (eg, Jan 5). But not necessarily the same year.
3/

What if it was a "40 under 40" list?

Or a "50 under 50" list?

Or in general: if we put M people on a list, what are the chances that *some* 2 of them will share the same birthday?
4/

Clearly, this is an exercise in probability.

To solve it, we'll assume 3 things:

1. No Feb 29 birthdays,

2. Each person on our list is *equally likely* to be born on any one of the other 365 days (Jan 1 to Dec 31), and

3. The birthdays are all independent of each other.
5/

Another way to state the problem:

We have M people, and a 365-sided fair die.

Each person is allowed to roll the die once -- and is thereby assigned a number between 1 and 365 (both inclusive).

What are the chances that *some* 2 people will get assigned the same number?
6/

Clearly, as the number of people (M) increases, so does the likelihood that *some* 2 of them will share the same birthday.

For example, suppose we have just 2 people on our list. That is, M=2. There's only a "1 in 365" (~0.27%) chance that they'll share the same birthday.
7/

But suppose we have 366 people (ie, M = 366).

Clearly, they can't *all* have different birthdays. There are only 365 days to go around. (Remember: no Feb 29 birthdays.)

So, there's a 100% chance that *some* 2 of them will share the same birthday.
8/

So, as M goes from 2 to 366, our probability of encountering "birthday buddies" goes from ~0.27% to 100%.

At what point do you think the probability crosses 50%? 75%? 90%? 99%?
9/

When asked questions like this, most people's first reaction is to *think linearly*.

At M = 2, the probability of birthday buddies is ~0%. By the time M = 366, it's 100%.

So the 50% mark should be crossed roughly halfway between 2 and 366, right? Say, at M = 180 or so?
10/

The right answer, it turns out, is just M = 23.

We need just 23 people on the list to give us a more than 50% chance of encountering birthday buddies.

That's the Birthday Paradox.

Our intuition, based on *linear thinking*, often misguides us in probabilistic settings.
11/

Charlie Munger's "Invert, Always Invert" mantra comes in handy when analyzing the birthday paradox.

Instead of asking "what's the probability of encountering birthday buddies", it's *much* easier to work out the probability of *not* encountering them.
12/

It's quite simple. If we *don't* want birthday buddies, we have to hope that *all* M people on our list have different birthdays.

This is like rolling a 365-sided die M times, and getting a different number each time.

Here's the number of ways that can happen:
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And there are 365^M total ways to assign birthdays:
14/

Since all these ways are equally likely, we can just divide one by the other to get the probability of *not* seeing birthday buddies:
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And we just invert this to get the probability of seeing at least one pair of birthday buddies:
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With this formula, we can plot the probability of seeing birthday buddies vs M.

From the plot, we see that as M increases, birthday buddies *rapidly* become more and more likely.

Linear thinking grossly *underestimates* this rapidity.
17/

To give you an idea of this rapidity:

At M = 23, the probability of seeing birthday buddies crosses the 50% mark.

At M = 32, the 75% mark.

At M = 41, the 90% mark.

And at M = 57, the 99% mark.
18/

We also gain insight by looking at the *incremental* contribution of each increase in M.

For example, when M = 10, the probability of birthday buddies is about ~11.69%. At M = 11, it's ~14.11%.

So, the 11'th person's *incremental* contribution is 14.11 - 11.69 = ~2.42%.
19/

Here's a plot of these *incremental* contributions vs M.

This plot is very interesting.

It shows that initially, there's a *law of increasing returns*: each increment to M produces progressively *bigger* increments to birthday buddy likelihood.
20/

But then, at around M = 20, this reverses course and becomes a *law of diminishing returns* instead.

Now, each increment to M produces progressively *smaller* increments to birthday buddy likelihood.
21/

In other words, up to M = 20, each person contributes *more* than the previous one.

But starting at M = 21, each person contributes *less*.

The 20'th person contributes more than the 19'th. But the 21'st person contributes less than the 20'th.
22/

This kind of "increasing returns up to a point, followed by diminishing returns after that point", is a common feature we see in many life situations.

It applies to learning new subjects. Building muscle. Returns on invested capital in many businesses.
23/

These situations are characterized by an "S curve".

Every S curve has an *inflection point*. This is where it transitions from increasing to diminishing returns. In our birthday paradox, this is M = 20.

When we see an S curve, it usually pays to think *non-linearly*.
24/

Here's a picture to help you think non-linearly.

As the picture shows, the key idea is to think in terms of *incremental* returns: are they increasing (convex), diminishing (concave), constant (linear), or at first increasing but later on diminishing (S curve)?
25/

There are at least 5 key lessons we can learn from the birthday paradox.

Key lesson 1: Simplify the problem to its essentials.

For example, we decided to ignore Feb 29 birthdays. This helped us get rid of many messy corner cases -- *without* causing us to lose any insight.
26/

Key lesson 2: Don't over-simplify.

Linear thinking is an example of over-simplification in this case. It causes us to dramatically underestimate the likelihood of seeing birthday buddies -- and thereby miss crucial insights.
27/

Key lesson 3: Think probabilistically.

Most outcomes in life are not deterministic. Chance often plays a big role.

So, it's usually a good idea to enumerate the various possible outcomes, work out which ones are desirable and undesirable, the odds of each, etc.
28/

Key lesson 4: Invert, always invert.

In many probabilistic situations, inverting the problem (eg, asking how many ways birthday buddies *cannot* occur) can help us solve it.

As Charlie Munger is fond of saying: I only want to know where I'll die, so I'll never go there.
29/

Key lesson 5: Think non-linearly.

This often means thinking in terms of *incremental* or *marginal* returns.

For this, it's useful to bear in mind mental models like convexity, concavity, S curves, inflection points, etc.
30/

As usual, I'll leave you with some useful references.

I love Shannon's 1952 speech outlining 6 methods for thinking creatively and solving problems strategically. Two of the methods are "simplifying" and "inverting". (h/t @jimmyasoni)

For more: https://t.co/QlNo5LAFzJ
31/

I also recommend listening to this (~1 hr, 23 min) podcast episode, where @ShaneAParrish and @Scott_E_Page discuss several mental models for both non-linear and probabilistic thinking -- including convexity and concavity, Markov chains, etc. https://t.co/tMHOojdePf
32/

Also, this article by @eugenewei on how to anticipate inflection points in S curves (he calls them invisible asymptotes) is excellent: https://t.co/J4IrhQz5zQ
33/

Finally, I want to thank my friend @SahilBloom.

It was his 30'th birthday earlier this week (and @aryamanar99's suggestion that I "gift" him a thread) that prompted me to reflect on birthdays and the birthday paradox.

Happy birthday, Sahil!
34/

If you're still with me, kudos to your perseverance!

Forget *non-linear* thinking. Most people can't follow a thread linearly from start to finish. But you're not one of them, and I appreciate it!

Take care. Enjoy your weekend!

/End

More from 10-K Diver

1/

Get a cup of coffee.

In this thread, let's talk snowballs.

Snowballs are super fun! And they can teach us so much about life, about things that grow over time, their rates of growth, compounding, etc.


2/

Snowballs are often used as a metaphor for compounding.

A snowball starts small at the top of a hill. As it rolls downhill, it picks up speed and grows in size. This is like money compounding over time.

For example, here's Buffett's famous "snowball quote":


3/

There's even a famous book about Buffett with "snowball" in the title.

The book's theme is similar to the quote above: the process of compounding is like a snowball that grows over time as it rolls downhill.

Link:
https://t.co/L3opOrdeoZ


4/

Clearly, snowballs rolling downhill are worthy objects of study.

So let's dive into their physics!

Luckily for us, in 2019, Scott Rubin published a paper analyzing such snowballs -- in a journal called "The Physics Teacher".

All we need to do is understand this paper.


5/

We begin by identifying 2 kinds of quantities in our "snowball system":

1. "Parameters" that don't change with time (eg, the hill's angle of incline), and

2. "State Variables" that *do* change with time (eg, the snowball's radius and velocity).

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Always. No, your company is not an exception.

A tactic I don’t appreciate at all because of how unfairly it penalizes low-leverage, junior employees, and those loyal enough not to question it, but that’s negotiation for you after all. Weaponized information asymmetry.

Listen to Aditya


And by the way, you should never be worried that an offer would be withdrawn if you politely negotiate.

I have seen this happen *extremely* rarely, mostly to women, and anyway is a giant red flag. It suggests you probably didn’t want to work there.

You wish there was no negotiating so it would all be more fair? I feel you, but it’s not happening.

Instead, negotiate hard, use your privilege, and then go and share numbers with your underrepresented and underpaid colleagues. […]

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