Chapter 6: Probability in Physics – Chance & Uncertainty

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Welcome back to the Deep Dive.

Today we are not talking about predictive analytics or the stock market.

Nope.

We are diving into something, well, maybe even more fundamental.

A huge shift in how science sees the world, laid out beautifully by Richard Feynman.

That's right.

We're digging into chapter six of his lectures on physics.

It's called probability.

Yeah, I mean, probability sounds almost simple, right?

Like coin flips and dice rolls.

It does, but Feynman takes it way beyond that.

He's not just talking about it as a tool for when we're, you know, ignorant about the details.

He actually frames it as the core logic of the physical world.

He even quotes Maxwell calling it the true logic of the world.

Okay, wow.

That definitely sets the stage higher than just

gambling odds.

So our mission today is to follow Feynman and see how physics takes this everyday idea of chance of likelihood and turns it into something rigorous.

Something quantitative.

Exactly.

A framework that's apparently essential for understanding.

Yeah.

Well, everything from how molecules move to where an electron might be.

We need to figure out how the universe itself sort of calculates odds.

And Feynman starts right at the beginning.

Why do we even need probability?

Why guess?

Because we don't know everything.

Basically, yeah, it comes down to incomplete knowledge, whether it's predicting a roulette wheel or, you know, the weather next week, or even exactly when a radioactive atom will decay.

We're always dealing with systems where we just have all the initial data perfectly pinned down.

Right.

So to make that guessing useful,

physics needs a solid definition.

You can't just say it feels likely.

Precisely.

And the key thing for making it quantitative in physics is that the situation has to be, at least in principle, repeatable.

Repeatable.

Okay.

Yeah.

So the definition becomes pretty straightforward then.

The probability of some outcome A, let's call it P A, is just the fraction of times you actually that outcome, N sub A, compared to the total number of times you run the whole observation, N.

So P A is just N A divided by N.

That's it.

P A equals N A over N.

But that repeatability thing is crucial.

If you can't repeat it, that ratio doesn't really mean much.

I think the example Feynman uses to make this concrete is really effective.

Taking it from just a ratio to a physical thing, the collision probability idea.

Oh yeah, the particle hitting the slab.

Let's picture that.

Okay.

Imagine you're firing a stream of particles, like little bullets, at a thin sheet of material.

A wall of atoms.

Right.

Mostly empty space, but with tiny targets inside.

Exactly.

Mostly space, but there are these dense little nuclei scattered around.

And the question is, what's the chance, the probability, that one of your incoming particles actually hits a nucleus?

So it stops being just about counting events and starts being about geometry.

Yeah.

Area.

It really does.

Because the nuclei are spread out, the chance of hitting one is basically like comparing how much area they block, their shattered area, to the total area of the whole slab you're firing at.

And physicists give that effective target area of one nucleus a name.

The cross -section.

Usually written with the Greek letter sigma.

Sigma, right.

So if you know how many atoms there are per unit area in your slab, let's say that's N divided by A, then the total probability of a collision, P subscript C, is roughly just that cross -section sigma times the density of atoms, N over A.

So key C is about sigma times NA.

It really connects it.

The abstract probability number comes directly from physical size and density.

Exactly.

It makes it measurable.

Okay, so that grounds the definition.

But then things get interesting when we think about actually doing the repeatable experiments.

Which brings us to fluctuations.

Right.

Like, okay, probability of heads is 0 .5.

Theory's clear.

But if you flip a coin, say, 30 times, you don't actually expect exactly 15 heads, do you?

Almost never.

That's the reality of randomness kicking in.

Even with a fixed probability, the results you observe, they bounce around.

They fluctuate.

The book describes this great test where they did exactly that.

Flipped a coin 30 times.

They called that one game.

Okay.

One game equals 30 flips.

Yep.

And then they repeated that entire game 90 times.

Wow.

Okay.

Lots of data.

Lots of data.

And sure, the most likely outcome for any single game is 15 heads right in the middle.

Yeah.

But the actual results showed a really significant spread around 15.

That spread is the key point, isn't it?

It wasn't all bunched up right at 15.

The graph they show, it's wide.

Very wide.

Things like 12 heads or 18 or 14 or 17.

Those results happened quite often.

Almost as often as 15 itself sometimes.

The distribution wasn't a sharp spike.

It was, well, spread out.

So there's a mathematical way to describe that spread.

It's not just random chaos.

Oh, absolutely.

It's described perfectly by what's called the binomial distribution, usually written PK.

And that formula calculates the exact probability of getting, say, K heads if you toss the coin N times in total.

How does it do that?

What goes into it?

The crucial ingredient is counting the number of different ways, the getting two heads and three flips.

You could have head to head tail or head tail head or tail head head.

Exactly.

And that's why the diagrams Feynman includes are so helpful.

The branching ones for like three tosses or six tosses.

Yeah, you can visually see the paths.

You see that getting three heads in a row, HHH, only has one path.

But getting two heads, like HHT, HGH, THH, has three paths.

So it's three times more likely.

Right.

And those number of ways are actually the binomial coefficients, you know, the numbers you find in Pascal's triangle.

Oh, okay.

Connected to that.

Yep.

For six tosses, the number of ways to get zero, one, two, three, four, five, or six heads are one, six, 15, 20, 15, six, one.

Ah, symmetrical.

And 20 is the peak in the middle.

Right in the middle.

Getting three heads is 20 times more likely than getting zero heads or six heads, just because there are 20 different sequences that result in three heads.

So the distribution naturally peaks in the middle and spreads out like a bell curve because that's where most of the possible paths end up.

Precisely.

The binomial distribution predicts that spread perfectly.

The fluctuation isn't some messy error.

It's the direct consequence of counting possibilities.

Okay, that makes sense.

Now, let's apply this idea of randomness to motion, the random walk.

Ah, yes, a classic model.

And it's super important, right, for understanding things like how molecules jiggle around Brownian motion or even how errors might add up in measurements.

Absolutely.

So the setup is simple.

Imagine a sort of player taking steps, n steps in total.

Okay.

And each step is random, either one step forward plus one or one step backward, max of one, with equal probability, like flocking a coin for direction.

Got it.

So the big question is, after n steps, how far away does the walker actually get?

Right.

What's the typical distance traveled?

Now, if you just calculate the average net distance, let's call it angle brackets d sub n.

Yeah.

What would you expect?

Well, since forward and backward are equally likely, you'd think they'd cancel out on average.

So zero.

Exactly.

The average net distance is expected to be zero over many, many trials.

But wait, the walker definitely moved.

They didn't just stand still.

Zero feels wrong.

It's incredibly misleading.

Like you said, the walker took n steps.

They went somewhere.

A zero average hides the actual journey.

It's like walking 10 steps forward and 10 steps back, net distance zero, but you walked 20 steps.

Okay.

So we need a better way to measure progress, something that doesn't let the pluses and minuses just wipe each other out.

We do.

We need something that cares about the magnitude of the displacement, not just the direction.

And what does that measure?

It turns out to be the expected value of the square of the distance written as angle brackets d squared, lancel d two wrinkle.

D squared.

Okay.

So squaring it makes both forward plus one squared is one and backward, angle one squared is one, contribute positively.

That's the trick.

Squaring eliminates the cancellation.

Every step, no matter the direction, adds to this squared distance measure.

Clever.

And what does the math show for this lancel d two wrinkle?

Well, Feynman walks through the derivation and it's surprisingly neat.

The expected value of the square of the distance after n steps, lancel d two wrinkle turns out to be simply n.

Just n, the number of steps.

Just n, lancel d two wrinkle, n no lar.

Wow.

So if we want a measure of the typical distance itself, not the square distance, we take the square root.

That's called the root mean square distance or DRMs.

That would be.

The square root of n.

DRMs equals the square root of n.

That's a huge result.

It's fundamental.

It tells us that the typical distance a random walker travels grows only as square root of the number of steps, not linearly with the number of steps.

That feels inefficient somehow.

Like to double your distance, you have to take four times as many steps.

Exactly.

Progress against randomness is slow.

The square relationship governs it.

And does this Courton idea connect back to the coin flip fluctuations we saw earlier?

Perfectly.

The expected amount of deviation in the number of heads, none h -adels, away from that perfect 50 % average.

That deviation also scales with Specifically, it's around plus or minus a square root of n.

Ah, so for our 30 flip game, scored n is about five point something.

About five point five.

Yeah.

So the standard deviation, the typical spread around the average of 15 heads is proportional to that five point five.

So the width of that bell curve we talked about, the fluctuation size, it's not arbitrary.

It's set by the square root of the number of trials.

Precisely.

And notice something interesting.

As n gets very large, the absolute deviation gets larger.

Right.

The range of likely outcomes widens in terms of raw numbers.

But the fractional deviation, which is like n divided by n, that actually goes to zero as n increases.

Ah, because scored n's are like one dollar square.

Exactly.

So with more and more flips, the fraction of heads gets closer and closer to point five, even though the absolute number of heads might be further from n too.

Okay, that clarifies things.

So even within randomness, there's this predictable scaling of the error or fluctuation.

It's one of the most powerful statistical relationships there is.

Okay.

So we've dealt with discreet things, head stales counts, forward, backward steps, but physics often deals with continuous quantities, right?

Like the velocity of a gas molecule, which could be anything, not just plus one or natural.

That's the next big step.

Moving from discrete probability counting things to continuous probability measuring things.

And this requires a new tool.

Yes.

We need the concept of a probability density function, usually written as PHX.

Okay.

Probability density.

What does that mean?

Well, for a continuous variable like position six dollars, the probability of finding the particle at exactly one precise point six dollars is basically zero.

There are infinitely many points.

Right.

So instead we talk about the probability of finding the particle within a small range or interval, let's say between six dollars plus delta X.

Okay.

A little window.

Exactly.

And that probability is given by PX at times the width of the window, delta X.

So delta X is the probability.

So PX itself isn't a probability, but probability per unit of X, like probability per meter or per meter per second.

You got it.

It's a density.

To get an actual probability, you have to multiply it by a small interval, delta X times or integrate PX over a larger range.

Okay.

And what does this density function PX look like for something like the random walk?

If we let the number of steps N get really, really large.

Ah, well, it settles into perhaps the most famous and important distribution in all of science.

Which is?

The Gaussian distribution, also known as the normal distribution.

The bell curve.

The classic bell curve shape.

Mathematically,

PX turns out to be proportional to DeLerre raised to the power of negative squared divided by two times sigma squared, propped to E by two.

Okay.

That E to the minus X squared shape and the sigma there, is that related to the spread again?

It absolutely is.

Sigma is the standard deviation and it determines the width of the bell curve.

For the random walk, sigma squared is proportional to N, the number of steps.

Which makes sense because we found the distance darkers grows as squirt no.

So the distribution should get wider as N increases.

Exactly.

As N grows, the bell curve spreads out, becoming wider and lower because the walker is more likely to be found further from the start.

But the total area under the curve always stays.

One.

Always one.

Because the probability of finding the walker somewhere has to be 100%.

Right.

And this Gaussian isn't just for random walks, is it?

Oh no, it's ubiquitous.

But more relevantly here, it's the kind of distribution you see describing things like the velocities of molecules in a gas.

That's Maxwell's distribution.

Knowing that distribution function allows physicists to calculate incredibly useful things, like the expected number of molecules that have speeds within a certain range, which is fundamental to understanding temperature and pressure.

So probability density functions become the language for statistical mechanics, for describing these huge systems of particles.

Precisely.

Okay.

So far, probability seems like a really clever tool for dealing with complex situations where we lack perfect information or where there are just too many parts to track.

Right.

It's been about handling complexity or ignorance.

But then we get to section 6 -5, the uncertainty principle.

And things change fundamentally.

They really do.

This is where probability stops being just a tool for us and becomes seemingly inherent to nature itself.

How so?

In classical physics, the idea is always, if we just knew the initial position and velocity of everything perfectly, we could predict the future perfectly.

Probability is just patching over our lack of perfect data.

Our ignorance, yeah.

But quantum mechanics comes along and says, nope, there are fundamental limits to how well you can know certain pairs of properties simultaneously.

It's not about building better instruments.

It's a law of nature.

And the famous pair is position and momentum.

That's the canonical example, Heisenberg's uncertainty principle.

It states that the uncertainty you have in a particle's position, let's call it delta x, times the uncertainty you have in its momentum, delta p, that product can't be zero.

It can't be arbitrarily small.

There's a minimum limit to the combined uncertainty.

Exactly.

The quantitative statement is that delta x times delta p has to be greater than or roughly equal to a fundamental constant related to Planck's constant, usually written as such a bar, delta x delta p chestburn bar.

So if you try to measure position really accurately, making delta x tiny,

then the uncertainty in momentum, delta p, must become large to compensate.

The probability distribution for momentum spreads out.

And vice versa.

It's a trade -off enforced by physics itself.

Yes, not by our clumsiness.

And this means for things like an electron and an atom, we can't talk about it having a definite position and momentum at the same time.

So how do we talk about it?

We have to talk about probabilities.

The state of the electron is described by a probability distribution.

Which leads to that amazing imagery Feynman uses,

the probability cloud for the electron in a hydrogen atom.

Isn't that a great picture?

Figure 611, it's not a little planet orbiting a sun.

No, it's fuzzy.

It's a cloud.

And the density of the cloud, how white it looks in the diagram at different places, that represents the probability density, p6 -wurgles, of finding the electron at that location.

So where the cloud is thickest, that's where you're most likely to find the electron, if you were to measure its position.

Exactly.

You can only state the chance of finding it here or there.

Its position isn't just unknown, it's intrinsically probabilistic.

Wow.

So probability moves from being a description of our knowledge to being a description of reality itself at that scale.

That seems to be the implication.

At the quantum level, probability is the language.

It's not a workaround, it is the description.

That really wraps up the journey nicely.

We started with probability just as a way to count outcomes, like coin flips.

Yeah, simple fractions based on repetition.

Then we saw how it describes the predictable scale of fluctuations in random walks, using that Sione idea.

Right, linking it to the spread and results and the distance traveled.

Then it became the tool for continuous variables via probability density functions, essential for gases and statistical mechanics.

The Gaussian distribution, Maxwell's distribution.

Yeah.

And finally, with quantum mechanics, it becomes the fundamental description of reality for particles, the cloud, not the orbit.

It's quite an evolution in perspective within one chapter.

So if you're pulling this all together, what are the main takeaways Feynman wants us to grasp about probability in physics?

I think he highlights two crucial roles.

First, it's absolutely necessary for dealing with super complex systems with tons of particles, like gases and thermodynamics or statistical mechanics, where tracking every single piece is just impossible.

Okay, the complexity argument.

And second, it's the inherent language needed for describing atomic and subatomic events, where the laws of quantum mechanics dictate that uncertainty is fundamental, not just a result of our ignorance.

So it tackles both overwhelming complexity and fundamental indeterminacy.

That's a good way to put it.

Which means, for anyone studying physics, getting comfortable with quantitative probability isn't just about learning a mathematical tool.

No, it's about learning the language the universe seems to use, both for large collections and for its smallest parts.

You need it to understand how uncertainty itself is measured and behaves.

It really drives home Feynman's closing thought, doesn't it?

That we sort of have to accept that the fundamental laws might really be based on uncertainty.

It's a profound idea.

It leaves you wondering, if the building blocks operate probabilistically,

does that mean the deterministic world we think we see at our scale is just an emergent property, a very, very likely outcome of countless underlying chances?

A kind of statistical illusion built on quantum fuzziness.

That's definitely something to chew on.

Indeed.

Well, thank you for guiding us through that deep dive into the true logic of the world, according to Feynman.

My pleasure.

It's fascinating stuff.