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In which Hank discusses the basics of probability, the art of finding meaning in randomness. And it's there, it's all over, everywhere! Master these simple rules and you'll always know what's coming...except not really.

Thanks to Karen Kavett for the graphics!


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A Bunny
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Good morning, John, I need you to do me a favor: go and get a coin and flip it ten times.

Actually, everyone who's watching this video right now, pause it, go get a coin, and flip it ten times. Now every flip of every coin is going to be entirely random. But despite that fact, I know what just happened when all of you flipped your coins.

I don't know what happened to your coin, but I know what happened overall. And that is because I'm a psychic. Because I am omniscient.

Because I have a little camera installed in all of your rooms and I WATCH YOU AS YOU WATCH ME. Okay, not actually, it's because I know about probability. In fact, as this video will probably be viewed tens of thousands of times, I can guarantee you that right now there are a bunch of people who are like "holy crap man, I just flipped a coin ten times and it landed on heads every single time.

I have a magical coin. A magical - I should take it to Vegas and bet on it because it always lands on heads." But guess what, my friend, there's nothing special about your coin and there's nothing special about you. I'm sorry, y- you- you're a beautiful snowflake, you're very special.

Even if the odds of getting ten heads in a row are over one in a thousand (which they are, they're 1 in 1024), assuming that there are 102,400 people who just flipped their coin ten times, there are roughly 100 people who just flipped ten heads in a row. And a hundred people who just flipped ten tails in a row. This is probability.

It is the science, the act, nay... the ART of assigning meaningful numbers to things that no one can actually really know. Probability lets us make real, predictable and measurable conclusions about random events. A weird thing about random events is that when they happen once you have no idea what's going to happen.

But oftentimes when they happen many times, patterns are obvious and unavoidable. So let's do a little terminology now. Probability is just P.

The event that we're looking for the probability of is expressed by a letter, usually starting with A. So the probability of A happening is P(A) ("P of A"). And if all outcomes are equally likely, P(A) equals the number of events where A is true, divided by the total number of events.

So when you all flipped your coins, each individual flip was entirely random. No one, not Miss Cleo, not Albert Einstein, not even Dr. Manhattan could know where that coin was going to fall.

But there are two possible outcomes and getting heads is one of those outcomes. So there is a 1 in 2, or one half, or zero-point-five or fifty-percent chance that it will land on heads. So when we flip a hundred coins roughly fifty of them are going to be heads and fifty are going to be tails.

And we expect some variation around those numbers. Okay, duh, obviously. Whatever we're curious what will happen when we flip two coins.

Then we're talking about the probability of A and B happening. Very simply we have a one in two chance twice. And that is 1/2 times 1/2, and assuming you know how to multiply fractions, that is 1/4.

A one in four chance that you will get heads and then heads. But then if we're doing it ten times we have 1/2 times itself ten times, which we can write as (1^2)^10 (one half to the tenth power). Which my calculator tells me is 1/1024.

This is how I could pretend to be so smart before. Now let's take things much more complicated and talk about dice instead of coins. It's not actually anymore complicated.

So if you're rolling a die, the total number of outcomes is six. And maybe we're concerned about whether or not you're going to get a five. That's one possible outcome.

That's 1/6. Holy crap, how crazy that you have a one-in-six chance of rolling a five. But what is the probability of it not being a five?

That, my friends, is called the opposite, or complement. And those of you who are actually thinking about that will probably recognize that that is 1-P(A). And the case of you not rolling a five is 1-(1/6) which is 5/6.

So the probability of not rolling a 5 is 5/6. This doesn't sound complicated but the idea of 1-P(A) being the opposite or complement is actually very important. But if you want to know the odds of the outcome being an even number then you have to take all the even numbers (two, four and six) and put them over the total number of possible outcomes.

And you have 3/6 which is 1/2. So basically, it's the flip of a coin. So there you have it.

The outcomes that you're concerned with over all the possible outcomes. But sometimes those two numbers are really hard to come by. Like it's really hard to figure out the number of possible poker hands is 2,598,960.

Unless you look it up on the internet, which is what I did, and that's probably what you should do in that situation, too. And what about the probability of getting two pair in your poker hand? That - how the heck would you figure that out?

Well you don't, you look it up on the internet. It's 123,552. So the odds of getting two pair in your poker hand is 123,552 divided by 2,598,960, which is 4.7%.

It's so easy! Because you're all so very smart, you probably have noticed that probabilities fall on a scale of zero-to-one, with zero being an impossible event and one being a certain event. And this is why probabilities are often represented in percents.

I assume that you know this, but percent is just like saying something is out of a hundred. So when something happens one out of two times the probability of 0.5 which is fifty out of a hundred times. So when we talk about rolling a die and figuring out what the probability of it landing on 1 twice is, we're talking about the probability of A and B happening.

Which is represented like this, and is basically like saying P(A) times P(B). Of course this is only the case when A is not dependent on what B is. So the probability of rolling snake eyes is 1/6 times 1/6 which is 1/36.

The probability of A or B happening is a completely different thing, and that's written like this and is basically P(A) plus P(B). So the probability of rolling a 2 or a 3 is 1/6 plus 1/6 which is 2/6 which is 1/3 which is a one-third probability. So now because you are theoretically going to have to use this new-found knowledge in actual life somehow, I'm going to make up some weird-ass situations that have nothing to do with real life and use them to test your knowledge of probability.

This is what we call "word problems." And when I was in school, I hated them and they made me want to kill myself. But now I'm teaching them so I frakkin' love'em. First word problem: Every time you smoke a cigarette, you have a one-in-five chance of being killed by a terrorist.

And every time you have sex before marriage, you have a 40% chance of being devoured by a tank of venomous piranhas. I think as long as I make situations up I should make up situations that also have value judgments and are at least a little bit interesting. So if you're not married and have a cigarette after you have sex, what are the chances that you survive?

I'm going to give you a little probability tip here: always go for "ANDs" and avoid "ORs." So if you're calculating the chance that we die it's either the terrorists or the piranhas that are gonna get us, and that's an OR, and ORs are bad. So make it into an AND, and say in order to survive we have to both survive the terrorist AND the piranhas. But in order to do that we have to do our 1-P(A) trick.

So we have our 1 minus 1/5 which is 80% and we have our 1 minus 40 is 60%, and we just multiply those two together and get 48%. Word problem number 2: You're at your mother's house and you've been shot by a poison dart by ninjas who've just cut the power. You grope around to find the medicine cabinet but your mother always mixes all the pills into one bottle, 'cause she thinks it saves space.

Which is dumb because there's a ton of space in the medicine cabinet anyway. It's dark so you can't see the difference between the pills, and inside the bottle there are 6 Tylenols, 4 Imodium ADs and 4 poison dart antidote pills. You only have to pick three pills out of the bottle before the poison dart kills you.

What are the odds that you survive? I think all word problems should be life or death. Okay, so for those of you who thought you were going to avoid this problem by just drinking the whole bottle all at once, you just DIED from poison dart antidote pill overdose.

You're DEAD and YOU CAN'T CHEAT YOUR WAY OUT OF WORD PROBLEMS. You're going to want to start this problem by saying, okay, I have a 1/14 chance the first time, a 1/13 chance the second time and a 1/12 chance the third time, but then you're doing OR again. It's either this pill OR this pill OR this pill.

What you want is AND. So you have to calculate your chance of not getting an antidote pill. So your chances of not getting an antidote pill are 10/14 times 9/13 times 8/12.

You get 33%, and then you just switch back using 1-P(A) and it's 67% odds of surviving. Final word problem: You have a gun. There's only one bullet in it, but you don't know which chamber it's in.

You're being attacked by a terrorist velociraptor. If you do manage to shoot the velociraptor there's only a 30% that it will stop before it kills you. You only have time to pull the trigger four times.

What are the chances that you survive? Again, we must start by calculating the odds that we do not fire a bullet. Five out of six chambers are empty. *click* Four out of five chambers are empty. *click* Three of four chambers are empty. *click* Two of three chambers are empty. *click* Multiply all those together to get the odds that you aren't going to shoot a bullet.

That's 32% and 1 minus 32% is 68%, the odds that you will fire a bullet. Then you multiply that by 30% to get the odds that you will actually stop the velociraptor. Your chances of surviving are 20%.

Thank you for watching my video. John, I look forward to watching you educate me more on Wednesday.