Chapter 4: Probability

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How often do you find yourself just trying to make sense of the, well, the unpredictable world around us?

We're constantly trying to gauge what might happen next.

Yeah, it's a natural human thing, right?

Trying to put a number on uncertainty.

And that's exactly what we're doing today.

Welcome to the Deep Dive, where we take complex info, break it down, make it understandable,

actionable,

really tailor -made for you.

And our mission today is all about probability.

We're digging into the core ideas, drawing from Mario Triola's elementary statistics to show you how it all connects to real world decisions.

By the end of this, you'll have a much clearer grasp on probability values, what they mean, and why they're like the absolute foundation for inferential statistics.

Yeah, get ready for some moments that might, you know, challenge your intuition a bit.

Okay, so let's start right there.

Why is probability such a big deal in statistics?

Sometimes it feels a bit like a separate math topic.

Oh, it's anything but separate.

It's really the bedrock for what comes later, especially something like hypothesis testing.

Hypothesis testing, right?

Test your eight stuff.

Exactly.

Probability helps us figure out if what we're seeing is genuinely unusual or if it could have just happened by, you know, random chance.

It's how we decide if results are statistically significant.

Okay, significant.

So what does that look like in terms of the actual probability numbers?

Well, probabilities always live between zero and one.

Zero means impossible.

One means certain.

What we're often looking for in statistics are those really small probabilities.

A tiny value, maybe like .01, signals an unlikely event.

A big one, like .99,

means it's very likely.

And this ties into the rare event rule for inferential statistics.

Sounds important.

It really is.

It's a core idea.

The rule basically says if you observe something and the probability of that specific thing happening is incredibly small under a certain assumption, then you should probably conclude that your initial assumption wasn't correct.

Okay, okay.

That needs an example, I think.

Absolutely.

Let's take the XOR gender selection method mentioned in the book.

In one study, they had 945 births and 879 were girls.

Wow, that's a lot of girls.

It is.

Now, the assumption we test is, what if the method has no effect?

What if it's just chance, like flipping a coin?

Okay.

Well, the probability of getting 879 or more girls purely by chance in 945 births is, well, it's astronomically small, practically zero.

So tiny you can't even really imagine it.

Exactly.

So the rare event rule kicks in.

Because that outcome is so unlikely under the no effect assumption, we reject that assumption.

Meaning we conclude that getting 879 girls wasn't just random luck.

It's a significantly high number, which suggests the XOR to me method actually does seem to have an effect.

It appears effective.

So if the chance probability is minuscule, we lean towards thinking it's not chance.

That tracks.

But you mentioned a caution.

Significance versus importance.

Yes.

Crucial point.

Statistical significance just tells you it's unlikely to be random.

It doesn't automatically mean the result is practically important or meaningful in the real world.

You always need context.

Don't mix them up.

Got it.

Okay, let's lock down some basic terms.

What are the absolute building blocks here?

Four key things.

First, a procedure.

Any process that gives you results, like rolling a die or even just a single birth.

Then an event.

Basically any collection of results.

It could be simple or complex.

Hence a simple event.

That's an outcome you can't break down any further, like getting a girl in one single birth.

It's just that one outcome.

So the last one.

The sample space.

This is super important.

It's the complete list of all possible simple events for a procedure.

For example, if the procedure is having three births, the sample space is all eight possibilities.

So it's like the universe of all possible basic outcomes.

Precisely.

You need to know that universe to calculate probabilities correctly.

Okay, so how do we calculate these probabilities?

Are there different methods?

We use P A for notation, right?

Probability of A.

Yep, P A means probability of event A.

And yes, there are three common approaches.

The relative frequency approximation.

This is based on observation, on data.

You basically run an experiment or observe a process many, many times.

Then P A is just the number of times A actually happened divided by the total number of times you ran the procedure.

Think $39 ,000.

It's based on actual history.

And you can't just say P crashes .5 because a flight either crashes or it doesn't.

Exactly.

That's where this method is essential.

And it leads to the law of large numbers.

Which says?

As you repeat the procedure more and more times, the relative frequency you calculate gets closer and closer to the actual true probability.

But big caution here.

This only works for large numbers.

It doesn't mean if you've lost 10 coin flips, you're due for a win on the next one.

Precisely.

That's the gambler's fallacy.

Each flip is independent.

The law of large numbers applies to the long run, not the next specific event.

OK, method two.

The classical approach.

This one requires a crucial condition.

All the simple events in the sample space must be equally likely.

Like rolling a fair die or flipping a fair coin.

Exactly.

If they are equally likely, then P A is the number of ways event A can happen divided by the total number of different simple events.

Remember the three births example?

Top eight outcomes.

Right.

If we assume boys and girls are equally likely, which isn't quite true, but close enough for this example, then the probability of getting three children of the same gender is two PBA divided by the eight total outcomes.

So 28 or 0 .25.

The key is confirming they're equally likely first.

You can't just assume passing a stats test has a 0 .5 probability.

Absolutely not.

Don't make that assumption unless it's justified.

Which brings us to the third approach.

Subjective probabilities.

What's your guess?

It's an informed guess, let's say.

It's when you use your knowledge of the situation to estimate the probability.

Like you might estimate your probability of getting an A in this course based on your study habits, past performance, etc.

It's subjective, not based on repeated trials or equally likely outcomes.

Gotcha.

And briefly, simulations.

Yeah, sometimes none of these work well and we use simulations to estimate probabilities.

We'll touch on that more later.

It's in section four to five.

OK, and rounding.

How precise do we need to be?

Good question.

Generally, give the exact fraction or decimal if you can.

If you have to round a final decimal answer, the rule of thumb is three significant digits.

So 13 becomes 0 .333.

But something like 0 .25 is exact.

You don't need to write 0 .250.

And use decimals, not percentages, for calculations in professional work.

Yes, stick to decimals between zero and one.

It avoids confusion in formulas.

Alright, let's talk about complementary events.

This seems like a useful shortcut.

A prime.

A prime or sometimes A with a bar over it.

It just means not A.

The complement of event A is everything in the sample space that is an A.

And the rule is super handy.

The probability of A happening plus the probability of A not happening must equal one.

So P A plus P A do one.

Which means if you know one, you can find the other instantly.

P A is E O 1 P A.

Exactly.

If, say, 89 percent of adults use the internet, peer end point 890, then the probability that a randomly chosen adult doesn't use the internet is simply 1 minus .890, which is 8 .110.

Saves a lot of work sometimes.

But again, gotta be careful.

Don't just divide numbers randomly.

Make sure you know the total number of items first, like in that ghost survey example.

Right.

You need the total number surveyed, yes answers plus no answers, to form the denominator correctly.

Don't just use the yes or no counts alone.

It's fascinating how these probabilities get interpreted in high stakes areas, like the FAA defining probable or improbable.

Yeah, those aren't just casual terms for them.

A probability of .000000001 or greater per flight hour is probable, they expect it might happen.

Less than that, down to .000000000001 is improbable.

And below that is extremely improbable, basically negligible.

It drives safety engineering.

It really grounds the numbers.

And it makes sense of things like lottery odds, mega millions, one in nearly 259 million.

Yeah, the analogy of picking one specific quarter coin from a stack 282 miles high really drives it home.

It's just an enormous amount of luck required.

Which brings us to gambling odds.

What's the difference between actual odds and payoff odds?

Super important if you're near a casino or racetrack.

Actual odds against an event A is the ratio P .A.

to P .A.

So probability of it not happening versus probability of it happening.

Payoff odds are set by the house, casino, whatever.

It's the ratio of the net profit you win to the amount you bet.

They are usually not the same as the actual odds.

How so?

Using roulette.

Perfect example.

There are 38 slots, 000136.

The probability of hitting number 13 is 138.

The probability of not hitting 13 is 3738.

So the actual odds against hitting 13 are 3738, which simplifies to 37 to 1.

But the casino doesn't pay 37 to 1, right?

They typically pay 35 to 1.

So if you bet $5 and win, they give you back your $5 plus

$175 profit.

$35.

If the odds were fair, 37 .1, they should give you $185 profit.

$375.

That $10 difference on a $5 bet is their built -in profit margin.

That's exactly how they stay in business.

They pay less than the true odds warrant.

And just to wrap this section, our intuition really fails us sometimes with probability, doesn't it?

Yeah.

The Caesar's breath molecule.

The birthday problem.

Oh, massively.

The birthday problem is a classic.

Most people are sharp that you only need 23 people in a room for a greater than 50 % chance that at least two share a birthday.

With 25 people, it's even higher.

Our gut feeling is just way off.

Okay, moving into section 4 -2.

We're often interested in combinations of events, right?

Compound events.

Yes.

A compound event combines two or more simple events.

The key words signaling these are typically or and and.

Let's start with or.

That suggests addition.

Generally, yes.

The addition rule is for finding P, A, or B, the probability that either A happens or B happens or they both happen.

How does it work?

Intuitively, you add up the ways A can happen and the ways B can happen, but crucially, you have to avoid double counting any outcomes that are in both A and B.

Ah, the overlap.

Exactly.

The formal rule handles this.

P, A, or B, plus P, B, P, A and B.

That subtraction term removes the overlap, so it's only counted once.

Like that drug testing example from the book.

Finding the probability of selecting someone who tested positive or uses drugs,

you add the tested positive group and the uses drugs group but subtract the ones who are in both categories.

Perfect.

You sum the relevant counts from the table, carefully avoiding that double count.

This naturally leads to disjoint events, sometimes called mutually exclusive.

Right.

Disjoint or mutually exclusive events are ones that simply cannot happen at the same time.

There's no overlap.

Selecting someone who is male and selecting someone who is female from the same group can't be both.

So for disjoint events, P, A, and B is just zero.

Correct, which simplifies the addition rule beautifully.

If A and B are disjoint, then P, A, or B, plus P, B.

And this connects back to complements too, right?

A and A prime are always disjoint.

Always.

An event either happens or it doesn't.

So K, A, or A, plus P, A, one, it all ties together.

Okay, what about and?

That points towards multiplication.

Typically, yes, especially when we're thinking about events happening in sequence or across multiple trials.

The multiplication rule helps us find P, A, and B.

And the formula involves conditional probability,

P, B, A.

Yes, the formal rule is P, A, and B equals P, B.

This P, B, A means the probability of B happening given that A has already happened.

So the probability of the second event might depend on what happened in the first event.

Exactly.

It's crucial to distinguish this and, A in trial one and B in trial two from the and in the addition rule context, which meant A and B both occurring in the same single trial.

This seems tied to independent versus dependent events.

It is.

Two events are independent if the outcome of the first event does not affect the probability of the second event, like two separate coin flips.

And depending.

If the outcome of the first does change the probability for the second, like drawing two cards from a deck without putting the first one back.

Because taking one card out changes the makeup of the deck for the second draw.

Precisely.

And be careful, dependence doesn't always mean one causes the other.

Think of two lights on the same faulty circuit.

They might fail dependently, but one didn't cause the other to fail.

This independence -dependence thing is really important when we're sampling from a population.

Huge implications.

If you sample with replacement, meaning you put each item back before picking the next, then the selections are independent.

Okay.

But if you sample without replacement, which is more common in surveys, the selections are technically dependent.

But calculating those changing probabilities can get really messy, especially with large populations.

It can.

Which leads to a very practical shortcut.

The 5 % guideline for cumbersome calculations.

Ah, right.

What's the rule?

If you're sampling without replacement, but your sample size is no more than 5 % of the total population size, you can basically pretend the selections are independent.

Treat them as independent to make the math way easier.

Why does that work?

Because if the population is huge compared to the sample, taking out a few items doesn't really change the overall probabilities in a meaningful way.

The effect is negligible.

Like sampling three employees out of 130 million for drug tests.

Technically dependent, but three is way less than 5 % of 130 million.

Exactly.

So you can just calculate P all three positive as P one positive, P one positive,

assuming independence.

It's a very useful approximation.

This probability stuff has real design implications too, like redundancy in engineering.

Absolutely.

Building an redundancy is a core principle for reliability.

Think airplane engines or backup hard drives.

Like the CJD hard drive example.

A single drive has maybe a 2 .89 % failure rate, so a 97 .1 % success rate.

Right.

P works equal 0 .971.

But if you install two independent drives, what's the probability that at least one of them works?

That's where compliments come in handy again, right?

It's one minus the probability that both fail.

Exactly.

P at least one works.

One P.

Drive one fails.

A and D drive two fails.

Since they're independent, that's 1 .02890 or 0 .0289.

Which calculates out to about 0 .999.

Yeah.

Almost one.

Look at that jump in reliability.

The chance of total data loss, both failing, drops dramatically by a factor of almost 29.

That's the power of redundancy.

Mathematically proven.

It also highlights again how careful you have to be with wording.

The birthday example, same day of the week, versus both born on Monday.

Huge difference.

P, same day, involves summing possibilities.

Mon -mon, two -two, etc.

Giving 17.

P, both Mondays, more specific.

Needing multiplication.

17 -17.

Giving 149.

Language is critical.

Let's revisit the chapter problem, too.

The X sort births.

P -20 girls in 20 births.

Assuming P -girls 12, using the multiplication rule for independent events, roughly.

It's 12, multiplied by itself 20 times.

12 -21.

Which is tiny.

Like 0 .0000954.

Extremely tiny.

Again, the rare event rule suggests this didn't happen by chance.

The method likely works.

Okay.

Section 4 -3.

Compliments, conditional probability, and Bayes' theorem.

Let's start with at least one.

We touched on this with redundancy.

Yes.

It's a common calculation, and the complement approach is usually the easiest.

Remember, the complement of at least one of something is none of that thing.

So P, at least one A, equals one P, no As.

Exactly.

Think about defective products.

If the defect rate is 15%, P defect equals 0 .15, then P not defective equals 0 .85.

If you buy 12 items, what's the probability at least one is defective?

Well, P, all 12 are good, would be 0 .85 multiplied by itself 12 times, assuming independence.

Right.

0 .8512, which is about 0 .142.

So P, at least one defective equals 1 .142 equals 0 .858.

Yep.

An 85 .8 % chance of getting at least one dud if the defect rate is 15%.

That's pretty high, likely unacceptable for quality control.

Definitely.

Now, conditional probability, PBA.

We saw this in the multiplication rule.

Yes.

PBA means the probability of event B occurring, given that you already know event A has occurred.

The given A part changes the calculation.

It restricts your focus.

How do we calculate it?

The intuitive way is often best.

Assume A happened.

Then, within that now reduced sample space, only the outcomes where A occurred, calculate the probability of B.

Like the drug test table again.

P positive test uses drugs.

We look only at the row or column representing uses drugs and find the proportion within that group who tested positive.

Exactly.

Out of the 50 users, 45 tested positive.

So 4550 equals 0 .900.

And P uses drugs positive test.

Now we look only at the positive test group.

Out of the 70 who tested positive, 45 actually use drugs.

So 4570 equals 0 .643.

Notice how different those are.

Hugely different.

And this is the confusion of the inverse.

That's the trap.

Thinking PBA is the same as TAB.

It almost never is.

The classic simple example.

P, it's dark.

It's midnight.

It's definitely dark if it's midnight.

But P, it's midnight.

It's dark.

That's almost zero.

It's dark for many hours that aren't midnight.

And confusing these has real consequences, like the prosecutor's fallacy in court cases.

Absolutely.

Confusing the probability of finding evidence, like a DNA match given innocence, with the probability of innocence given the evidence.

They are not the same.

And mixing them up can lead to wrongful conclusions.

It's a serious statistical error.

This seems like a good place for Bayes' Theorem.

What does it do?

Bayes' Theorem is essentially a formal way to update a probability based on new evidence.

It mathematically connects PBA and PAB.

So you start with an initial probability,

a prior probability.

Yes, your belief before the new info.

Then you incorporate the new information, like a test result, to calculate a revised probability, the posterior probability.

The medical test example seems perfect here.

Prior probability of cancer might be low, say 1%, PC 10 .0 .1.

Then you get a positive test result.

Right.

And the test isn't perfect.

It has a true positive rate, P positive cancer, maybe 80%, and a false positive rate, P positive no cancer, maybe 10%.

Bayes' Theorem, or just using a table like Table 4 -2, lets you calculate P cancer positive tests.

And the result is often counterintuitive.

Very much so.

With those numbers, P cancer positive test turns out to be only about 7 .5%.

Even though the test seems pretty good, 80 % true positive, a positive result doesn't make cancer highly probable because the initial prevalence was so low and false positives happen.

And you mentioned physicians often get this wrong.

Studies showed many estimated the probability at 70 -80 % after a positive test,

massively overestimating the risk due to that confusion of the inverse.

Using a simple table with hypothetical numbers makes the calculation much clearer and avoids the trap.

So Bayes helps refine probabilities with new data.

Coast Guard uses it for searches.

Yep.

They use Bayesian methods to update the probable location of someone lost at sea as search efforts continue and new information, or lack thereof, comes in.

It helps narrow the search area efficiently.

And it explains why coincidences aren't always so coincidental, like winning the lottery twice.

Exactly.

The probability of one specific person winning two specific lotteries is miniscule.

But the probability of someone, somewhere, winning some lottery twice over several years is actually much, much higher than our intuition suggests.

It's all about framing the probability question correctly.

Section 4 -4.

Counting.

Why is counting so important for probability?

Because often, to use the classical approach, outcomes total outcomes, you first need to figure out how many total outcomes there are and how many ways your specific event can occur.

And sometimes those numbers are huge.

So we need shortcuts.

Rule number one.

Green application counting rule.

If a procedure involves a sequence of steps or choices, you multiply the number of options for each step to get the total number of combined outcomes.

Like the password example.

5 characters, 92 options for each.

Total possibilities equals 92, 92, 92, 92,

or 925, which is over $6 .5 billion.

Your chance of guessing it on the first try is 1 divided by that massive number.

Shows why complexity matters.

Make sense.

Rule two.

Factorial rule n.

Factorial n is used when you're arranging n distinct items, using all of them.

And the order matters.

n, n1, n, n2, 1.

And importantly, 0 is defined as 1.

Example.

Arranging the letters in the word steam.

There are five different letters.

How many unique arrangements?

5, 5, 4, 3, 2, 1, equals 120.

The chance of randomly getting AEST, alphabetical order, is 1 out of 120.

Okay, so factorials are for arranging all distinct items when order matters.

But what if we're only selecting some items or some items are identical?

And this order thing, permutations versus combinations.

Exactly.

This is the crucial fork in the road.

Does the order of selection matter?

How do we remember which is which?

Permutations.

Order matters.

Think permutations position.

Different sequences of the same items count as different outcomes.

Combinations.

Order doesn't matter.

Think combinations committee.

Different sequences of the same items count as the same outcome.

Okay, let's do permutations first.

Order matters.

What if we're selecting our items from n different items, like the trifecta bat?

Right.

Selecting the first, second, and third place horses from a field of 20.

Order absolutely matters.

This uses the permutations rule, NPR.

The formula is n, nr.

For the trifecta, it's 20p3 equals 20, 2017.

20, 1918 equals 6 is 840.

So 6 ,840 different possible trifecta tickets.

Probability of winning randomly is 16 ,800, though horses aren't equally likely.

Correct.

Now there's another permutation rule.

What if some of the n items you're arranging are identical?

Like arranging the letters in Mississippi.

Exactly.

You have 11 letters total, but multiple i's, s's, and p's.

The formula becomes n divided by the factorials of the counts of each identical group.

So 11, 4, 4, 2.

This corrects for the overcounting caused by identical items.

The book uses a survey design example.

Right.

Ten questions, some identical.

It still results in a huge number.

Showing arranging even slightly similar items gives many possibilities.

The very large number, yes.

Okay.

Now the other path.

Combinations, NCR.

Order doesn't matter.

Right.

Selecting our items from n different items where the sequence of selection is irrelevant, like forming a committee or picking lottery numbers.

The Florida cash five lottery.

Yeah.

Pick five numbers from 35.

Order doesn't matter.

Exactly.

We use the combination formula, NCR, NNRRR, for the lottery 35C5 equals 35, which calculates to 324 ,632.

So one chance in 324 ,632 to win.

Still terrible odds.

Hence the phrase attacks on people who are bad at math.

And a tip.

If you play, avoid common patterns like 1, 2, 3, 4, 5, 6.

Because if you win, you'll likely have to share the jackpot with many others who pick the same obvious numbers.

Good practical advice.

Can we clarify the permutation versus combination difference one more time?

The corporate officers versus committee example.

Perfect contrast.

Scenario one.

Select three officers, CEO, chair, COO from eight candidates.

Does order matter?

Yes, absolutely.

Being CEO is different from being CEO.

Right.

So it's a permutation.

8P3 was eight.

83.

336 possible slates of officers.

Okay.

Scenario two.

Select a three person planning committee from the same eight candidates.

Does order matter?

No.

A committee of Alice, Bob, Carol is the exact same committee as Carol, Alice, Bob.

So it's a combination.

Yes.

8C3 was eight.

Eight to three.

Three O per eight.

If you six possible committees, look at the difference.

336 versus 56, just based on whether order counts.

That really drives it home.

And some fun counting facts wrap this up.

43 quintillion Rubik's cube positions, seven shuffles for a card deck.

Yeah.

And the surprising result for the random secretary problem, even with tons of letters, the probability of at least one getting into the right envelope stays around 0 .632.

Probability can be weird.

Okay.

Last section, four to five.

Simulations for hypothesis tests.

Why simulate?

Simulations are basically pretend versions of real processes.

They're super useful in two main ways, especially when direct calculation is hard or impossible.

Use number one.

To test a claim about a population parameter, like that body temperature example,

the claim or common belief is the average is 98 .6 degrees Fahrenheit.

But a sample of 106 people showed a mean of 98 .2 degrees Fahrenheit.

Right.

So we simulate, assuming the true mean really is 98 .6 degrees Fahrenheit, what kind of sample means would we typically get from samples of 106 people?

We use software to generate many, such samples randomly.

And then you see where the actual observed sample mean, 98 .2 degrees air falls compared to all the simulated means.

Exactly.

If 98 .2 degrees air is way out in the tail, far from the cluster of simulated means, which are centered around 98 .6, then it's highly unlikely to occur if 98 .6 were the true mean.

So using the rare event rule again, basically.

Precisely.

The simulation shows 98 .2 is significantly low, suggesting the 98 .6 assumption is probably wrong.

The actual probability is incredibly tiny, confirming the simulation's result.

Okay, that makes sense for testing claims.

What's the second use?

To find a probability that's just too darn difficult to calculate formally.

Like the shared birthday problem, but maybe with three people sharing.

Exactly.

Calculating P at least three share a birthday among a hundred people is mathematically horrendous.

But simulating it is easy.

Generate a hundred random birthdays, numbers one, 365,

sort them, check for triples, repeat thousands of times.

The proportion of simulations with a triple gives you a good estimate of the probability.

Yep.

Lots of software can do this.

Statisk, Statcrunch, Excel,

R.

Yep.

Many tools available.

And a real world example of simulations,

or maybe misuse, the lottery rigging case.

Ah, Eddie Tipton, the lottery security director who wrote code to influence the random number on specific dates.

He essentially ran simulations to figure out predictable patterns, drastically cutting his odds from millions to one down to maybe 200 to one for certain drawings.

He won millions before getting caught.

Shows the power and the danger, if not used ethically.

Definitely.

And a final critical point.

The simulation must accurately mimic the real process.

The dice example is key.

Simulating rolling two dice by just generating random numbers from two to twelve is totally wrong.

Because the sums when rolling two dice aren't equally likely.

A sum of seven is way more probable than a sum of two or twelve.

The simulation has to reflect the underlying probabilities of the actual procedure, or the results are garbage.

Got it.

The simulation has to behave like the real thing.

That's the key.

Wow.

Okay.

So we've covered a lot of ground.

Basic probability concepts, the rare event rule, how to calculate probabilities using relative frequency, classical, and subjective methods.

The addition and multiplication rules, the importance of complements, conditional probability, avoiding the confusion of the inverse, and how Bayes' theorem lets us update beliefs.

Plus all those essential counting techniques, multiplication rule, factorials, permutations, combinations, and finally, how simulations help us test claims and tackle tough probability questions.

It really forms the foundation for so much statistical reasoning.

Probability isn't just math theory.

It's a lens for looking at uncertainty everywhere.

Medical tests, system reliability, interpreting data.

It really equips you to think more critically about the numbers and claims you encounter every day.

To ask that crucial question, could this have just happened by chance?

Exactly.

And understanding that probability helps you make much more informed judgments.

So maybe a final thought for everyone listening.

The next time you see a surprising statistic or claim, pause and think about the probability.

How likely was this outcome just by random variation?

And how does that change how you view the claim?

That's a great takeaway.

Always question the likelihood.

Thanks for joining us on this deep dive.

Indeed.

Keep learning, keep questioning, and stay well informed.