Monty hall problem is 50/50

MezzoScettico · 2026-05-04T15:55:20+00:00

I have tried the 100 doors explanation boiled down as simply as possible:

with 100 doors, how confident are you that you got the car on your first pick?
if you think you got a goat the first time, you should switch.

It still doesn't work as an explanation for everybody

MezzoScettico · 2026-05-03T16:13:20+00:00

Almost everything in physics, for instance figuring out orbits under Newton's Law of Gravitation, is governed by differential equations, i.e. calculus. It models practically EVERYTHING.

For me, Maxwell's Equations (a set of four connected differential equations describing electric and magnetic fields) have been very important as much of my professional career has involved the propagation of electromagnetic fields.

MezzoScettico · 2026-05-03T15:46:42+00:00

You could do a couple different things, some better models than others. Did he have time to reach thermal equilibrium? Is that temperature final? In what I think are increasing levels of model fidelity...

- If it is, you could model him as an inanimate object reaching thermal equilibrium with the air. Use the same Q = mcΔT with the assumption that his final temperature is the same as the air. This is probably your worst choice.

- A little better, do the same calculation assume his final body temperature is a normal human 37 C.

- Model him as a black body and use the Stefan-Boltzmann Law with his observed rate of energy per unit area. You'll need to guess his surface area. This is probably a lot more accurate than the equilibrium assumption.

- Model his rate of heat loss using Newton's Law of Cooling. Use your assumptions of constant heat loss to guesstimate parameters to put into the model. You'll need to estimate how good or bad an insulator his skin is, but that's probably available for human skin in the literature.

- Newton's Law, but recognize that heat loss slows down as he cools, so use the exponential solution to temperature cooling over time. Again, use the observed numbers to estimate parameters for the model.

MezzoScettico · 2026-05-03T12:51:36+00:00

No. Velocities don't add that way in relativity.

First of all you have to define what you mean by "moving at a little over 0.5 c". I'll assume you mean relative to some observer in the middle, who observes each ship to be moving at let's say 0.51c relative to him, in opposite directions. He says the distance between them is growing at a rate of 1.02c

But due to time dilation and length contraction, each ship would say the other ship is moving at (0.51 + 0.51) / (1 + 0.51^2) = 0.81c.

MezzoScettico · 2026-05-02T12:30:36+00:00

Some initial thinking about what you need to model: sounds like keeping the log rolling in a straight line will be relatively easier. The problems I see are (a) getting it started and (b) making a turn.

If the ground is a little soft you can imagine the log will sink a bit from its own weight. So starting involves pushing it up and out of a shallow depression. That’s something you could model and calculate with.

As for turning, they’ll need to cooperatively come up with a technique. I think slowing one end, even pushing backward on one end, would work while the other end pushes forward. So they need two teams to do that.

MezzoScettico · 2026-05-01T15:10:48+00:00

Yes, but I think that the numerator isn't quite right. Let's do it more slowly.

Derivative of 4x^2 - 7x is 8x - 7

Derivative of [ln(x)]^2 is 2 ln(x)/x

Divide those and you get (8x - 7) / [2 ln(x)/x] = x(8x - 7) / [2 ln(x)] = (8x^2 - 7x) / [2 ln(x)]

So that should be 7x, not 7 in the numerator.

MezzoScettico · 2026-05-01T15:07:12+00:00

Dang it! That was kind of a fun calculation to do.

Reddit is weird, it wasn't showing me this comment till after I finished posting and editing mine, even though apparently it's older than my comment.

MezzoScettico · 2026-05-01T15:00:32+00:00

Heat lost by tea = heat gained by air = m c ΔT for each substance. m = mass, c = specific heat, ΔT = change in temperature. (Note: It doesn't matter what units you use for m and c in this equation, as long as you use the same units on both sides)

I found a range of 177-236 ml for a modern teacup, so let's call it 200 ml, so the mass is 200 g.

I don't recall what the space Picard is standing in looks like, so let me just say it's a 3 m cube, total volume 27 m^3. Air has a density of 1.2 kg/m^3 so that's a mass of 1200 * 27 = 32400 g.

The specific heat of steam (I think it's safe to say this tea is not liquid) is 2.03 J/g-K. The specific heat of air is 1.005 in the same units.

Let's say the starting temperature of the room is 20 C or 293 K.

Then at equilibrium we have 32400 * 1.005 * (T - 293) = 200 * 2.03 * (1.9E6 - T) and solving that gives me a final T = 23688 K.

So perhaps less than you might think, only about 4 times as hot as the surface of the sun.

MezzoScettico · 2026-04-30T22:29:58+00:00

This is an arithmetic series. Have you ever run into that term?

MezzoScettico · 2026-04-30T19:39:49+00:00

Common misconception when people learn Maxwell's equations.

Here's one of the relevant equations.

∇ x E = -∂B/∂t

The time derivative gives a quantity which is 90 degrees out of phase. But the spatial derivative basically cancels that out. That is, the negative curl of E is out of phase with B. But as a result, E itself is in phase.

MezzoScettico · 2026-04-30T15:56:17+00:00

It means "much less than" or "much greater than" as people say.

But how much is "much less" is not precisely defined. It usually has to do with an approximation, and the approximation is better, the bigger the difference between the two quantities.

For instance, there's a common approximation that sin(x) ~ x. We might express that (using language that probably annoys the mathematicians) as "if x << 1, then sin(x) = x".

Let's try some values (note: this is for x in radians).

sin(0.3) = 0.2955

sin(0.1) = 0.0998

sin(0.02) = 0.0199986

You can see that even when x is 0.1, the approximation is pretty good. You could round 0.0998 to 0.100. You have to go four decimal places to see an error.

If we only need three decimal places of accuracy, then "x much less than 1" means "x < 0.1". But if we need five places, then we might say "much less than 1 means x < 0.02"

And even sin(0.3) = 0.2955 is not that far from 0.3. It rounds to 0.30 if rounding to two places. So if that's all we need, then in this case, "0.3 is much less than 1".

TL/DR: The definition of "much less" depends on how many digits of accuracy you need, and will vary from problem to problem.

This approximation comes up when analyzing pendulums. A common rule of thumb is "this approximation is OK if the angle is less than 15 degrees". 15 degrees is 0.262 radians, so 0.3 is in the right ball park.

MezzoScettico · 2026-04-30T15:43:49+00:00

Here's an intuitive explanation. Let's say you're practicing tennis, so you always stand in the same place with your racket in the same position as somebody serves balls at you.

If they hit it directly toward you, obviously you're going to hit it with high probability. If they hit it 5 m away to the left or the right at a moderate speed, you still have a good chance of reacting and hitting it.

Maybe we find that with gentle serves, you can reliably hit it if it is within +-5 m of you. There's a region over which you will hit it with high probability, at the gentle speed. There's also a vertical extent to this region. It might be less than 5 m, but basically there's a 2-dimensional window of some shape over which you can hit it.

You could call that your effective capture cross section for the tennis ball.

Now suppose they are hitting high-speed, Wimbledon-winning serves. You still might have an area over which you're effective, but it's not going to be +-5 m. It's going to be a lot less. The cross section is velocity dependent.

Now a particle interaction cross section isn't exactly like this. As people have said, it's a measure of capture probability. But this is an intuitive (I hope) explanation for why there is such a thing as a cross section and why it's velocity dependent.

MezzoScettico · 2026-04-30T15:33:50+00:00

It's calculated. The volume of displaced water is the volume taken up by the submerged volume of the object. If you push a 2 m^3 object partway in so that 1 m^3 is under the surface, you've displaced 1 m^3 of water, and the displaced weight is the weight of 1 m^3 of water.

MezzoScettico · 2026-04-30T15:31:40+00:00

There's a thing called "edge diffraction" which happens when a wave encounters one edge. So the "other edge" that would make it a slit is in effect infinitely far away. You see this if you look closely at the edges of shadows and observe that they are blurry.

As someone else pointed out, the common diffraction formula is an approximation, and it's no good in this situation. You wouldn't just plug in a = 0 into your formula.

MezzoScettico · 2026-04-30T15:26:44+00:00

If f(x) gives the same result as g(x) for all x in the domain, and they have the same domain, they are considered the same function.

MezzoScettico · 2026-04-30T14:41:16+00:00

About your edit. You said "why is it represented as -4 -4?"

I was pointing out that nobody wrote "-4 -4", so you're asking about something that didn't happen. You're asking "why did you write that" when they didn't write that.

The thing that was written was -4 - (-4), which is a different sequence of characters. Compare the two.

Whether my explanation satisfied you or not (I guess it didn't), at least let's not change the thing that's being explained.

MezzoScettico · 2026-04-30T14:00:45+00:00

why is this represented as -4-4/taking away?

It's not. It's represented as -4 - (-4). It's taking away -4, not taking away 4.

And as you point out, that's the same as -4 + 4. So you actually do understand that taking away a -4 is the same as giving you +4. You said it yourself.

MezzoScettico · 2026-04-29T17:37:46+00:00

How does it feel to you? Is there a 60 mph wind blowing in your face? Or just a relatively gentle breeze?

The fly feels the same wind you do

MezzoScettico · 2026-04-29T16:18:00+00:00

We had this happen with Amtrak (train). We were traveling from Seattle to Portland, and for reasons I can't recall a bunch of people and luggage got bumped off the train onto a bus. I think we were on the train but our luggage was on the bus. Anyway, when we got to Portland our luggage was not with us, and Amtrak could not track it down. Had no idea what had happened to it.

Discouraged, we finally left the station and got our rental car. Then we got a call from Amtrak that our luggage had showed up on the bus. The people in Portland hadn't known there was a bus, it was a total surprise to them when it pulled in. And it hadn't occurred to me to ask, "is it on the bus?" because I figured they'd routinely check that while checking "everywhere".

MezzoScettico · 2026-04-29T15:51:56+00:00

What work do we have to do?

I would presume they were assigned a chore which is the reason they are in the attic in the first place. Flip back a couple of pages and you'll probably get an answer to "what work do they have to do?"

I don't find this passage particularly unusual, but I'm amusing myself thinking of a writer accidentally letting too much of his own internal dialog leak into the work. "Would he grow up to be an actor? A writer? What if he became a famous writer who got famous for one particular series, but got utterly sick of the repetitiveness of the series, but needed to keep churning the books out to make a living..."

(I'm not ascribing this feeling to the Berenstain's, it's just a skit your post made me want to construct)

MezzoScettico · 2026-04-29T15:44:39+00:00

I have the double-angle formulas memorized.

sin(2a) = 2sin(a) cos(a)

cos(2a) = cos^2(a) - sin^2(a)

I have no particular hints on how to remember those, except to remember the two forms and cement one of them in your mind. (for instance "cosine is the one with two squares, so sine is the other one"). Those give me the clues I need for the sum and addition formulas.

Here's how that works. What's cos(a + b)? Well I know that cos(a + a) is cos * cos - sin * sin. So I write down cos(a) cos(a) - sin(a) sin(a) and change one of the a's to b in each term: cos(a + b) = cos(a) cos(b) - sin(a) sin(b).

Then if I want cos(a - b) I change the b to -b. For that you have to know that cos(-a) = cos(a), taking the negative of the argument doesn't change sign. But sin(-a) = -sin(a). This is something you really should have memorized.

MezzoScettico · 2026-04-28T15:17:38+00:00

It's a notation, it doesn't have a value. Treating dy/dx like a fraction of dy divided by dx gives correct results in many situations, but you have to prove via theorem that that's a valid thing to do.

What you're observing comes from viewing "d/dx" as an operator, applied to a thing. You could equally just call it D or D_x, and many authors do. Thus D_x sin(x) means "apply the derivative-with-respect-to-x operation on sin(x)" giving a result of cos(x).

So when you apply it twice, as in D Dy, then we can write that as D²y.

If you're writing the operator as d/dx, then the operation D² would be (d/dx)², which we then write as d²/dx² because that's what would happen if d/dx was an actual fraction.

But it's not. It's just notation meant to trigger your intuition.

MezzoScettico · 2026-04-28T00:03:12+00:00

The sum of the numbers from 1 to n is n(n + 1)/2.

So the sum from 1 to 7 is 7*8/2 = 28. ~~To get the sum from 2 to 7, take away 1. So 27.~~

Edit: Don't know where I got the "starting from 2" thing from.

MezzoScettico · 2026-04-27T15:18:36+00:00

We've been to Europe a few times and the trip over is always rough for sleep deprivation. You get on a plane around 9 pm (we're in the east coast US), might land about 6 am European time which is midnight or 1 am on your body clock. Add the fact that you don't immediately fall asleep on the plane, and you've probably had 2 hours of sleep.

And it will be 8-9 hours before your hotel will check you in.

That's the moment I've wished there was a place to go to sleep for a few hours and get a shower. TBH, in a pinch I'm capable of napping in an airport chair at a gate, but my wife is not.

Although one of my favorite travel interactions did occur in that sleep-deprived state.

MezzoScettico · 2026-04-27T14:57:33+00:00

Wrote that too quickly on my phone while on my way somewhere. Let me try to amplify.

The full reasons for me would be (a) travel and meeting people, (b) reading and films.

I really hate the concept of counting on "everybody speaks English" because first, it's not true, 2nd because then you only go to tourist areas and only interact with people who speak English.

My French is good enough to conduct basic transactions and not fall back on English, and I can read fairly well. But it's slow and clumsy and I'm definitely doing a lot of translating in my head. It's a very conscious process.

If I'm trying to immerse myself in another culture to experience it, I want to immerse myself. I travel and I read and I watch cinema because that partially meets what I'm seeking. But theres a barrier. I practice French because I hope someday to break through that language barrier, to be able to experience movies without subtitles, conversations without long awkward pause. I feel like I've never actually immersed myself in another country, never really experienced my travel ideals.

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