Old school probability (very optional)
Probability (part 6) Introduction to conditional probability
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- Now our probability problems will get a little bit
- more interesting.
- So let's say I have a bag full of coins.
- So this is my bag of coins, maybe it's a bowl of coins.
- From a bunch of coins in there.
- I didn't need to draw it, but I did anyway.
- And let's say that I know that 9 of these coins are normal.
- So what do I mean my normal?
- Just, you know, let's say they're quarters and they have
- a heads and a tails, and that there's a 50% chance of getting
- a heads or 50% chance of a tails if you flip
- any of those coins.
- And then, 1 of the coins is a two-sided head coin.
- So essentially, no matter what you flip it, it's
- going to be a heads.
- So I have this bag, what I'm going to do is I'm going to
- blindfold myself, stick my hand in the bag, and
- pick a coin out.
- And then I'm going to flip that coin, I'm going
- to flip it five times.
- And my question to you is, what is the probability that
- I get all 5 to be heads?
- So whenever you see something unfamiliar it'll always work--
- maybe we'll do a tree or we'll do a hybrid of a probability
- tree or something else.
- But you say, well, at the very beginning of this experiment
- something happens.
- I'm going to pick a coin out of this bag.
- And so you could say that there's a-- well, how
- many coins are there?
- There's 10 coins and I have an equal probability of picking
- any one of the coins.
- Let's say I can't feel whether it's heads or tails.
- I'm doing it with gloves on, so I can't feel out
- the two-sided coin.
- Take my word for it, there's an equal probability to take
- any 1 of these 10 coins.
- There's 9 normal, 1 weird with 2 heads on it.
- So what's the probability that I get a normal coin?
- Well, there are 9 normal coins out of the 10 equally
- likely coins I could get.
- So the probability of a normal is 9 out of 10.
- So there's a 9 out of 10 chance that this is a normal coin
- Let's just call that normal.
- And then, what's the probability that I get
- a-- let's call it a 2s-- two-sided coin.
- I'll call it 2s.
- Well, that's just 1 out of 10.
- I'm either going to get this or that.
- I'm not going to take nothing out of the bag I assume.
- OK, so this is a 1/10 chance that I get a two-sided coin.
- And then whichever coin I get I'm going to flip
- it five times in a row.
- And I'm not going to do the tree here because
- to drew a tree I'd have 32 nodes on each side.
- But given this, given the normal, what is
- now the probability of getting 5 in a row?
- The way I'm going to write that is the probability of 5 heads--
- or I could have written 5 out of 5 heads, whatever.
- And then this upper line, this means given-- given that
- I have a normal coin.
- So if I can assume that I had a normal coin, what are the
- probabilities that my next 5-- so I'll say this is
- 5 out of 5 heads.
- Well, we figure that out.
- Each time there's a 1/2 probability and I'm going
- to do it 5 in a row.
- So it's 1/2 to the fifth, which equals 1/32.
- So the probability that I get 5 out of 5 heads given a
- normal coin is 1 out of 32.
- Fair enough.
- Well, what's the probability of 5 out of 5 heads,
- given the two-sided coin?
- And now just is this random notation I'm coming
- with the two-sided.
- But this upwards and downwards line, this vertical
- line, that means given.
- I mean I could flip this coin a million times and all of
- them are going to be heads.
- So the probability here is 1 out of 1.
- I'm definitely going to get a heads here.
- So all I asked initially is, what is the probability that
- I get 5 out of 5 heads?
- Well, this is going to be equal to-- and I'll switch colors
- arbitrarily here just to keep things interesting.
- This is going to be equal to the probability that you get 5
- out of 5 heads given-- let's call that a normal coin-- given
- a normal coin times the probability that you had a
- normal coin times the probability of
- let's say normal.
- Plus-- and I'm running a little out of space-- the probability
- that you get 5 out of 5 heads given the two-sided coin times
- the probability that you had the two-sided sided coin.
- Hopefully that makes sense and you kind of think of it in this
- way, what is the probability of getting to this outcome?
- It is the probability that you picked a normal coin and then,
- given the normal coin, there's a 1 in 32 chance
- that you got there.
- So it'd be 1 out of 32 times the chance you've
- got the normal coin.
- So 9 out of 10.
- And what's the probability of getting here?
- What's the probability you had to-- and, in general, in a
- tree, you just have to multiply all of the nodes to get to that
- point to figure out what's the probability of getting there.
- And I think if you watched the first few videos it make
- sense why that happens.
- So what's the probability of this?
- Well, there's a 1/10 chance that you picked the
- two-sided coin.
- And then of course, you're guaranteed to get all heads.
- This is 1 times 1/10.
- And all I did here and here is the same thing
- that I wrote up here.
- Let me switch colors.
- So what is this, the probability of 5 out of 5 heads
- given a normal coin times the probability of a normal coin?
- Well, that's just this.
- This is the probability of 5 out of 5 heads given the normal
- coin times the probability of a normal coin.
- And this is equal to what?
- 9/320.
- And then this, the probability of 5 out of 5 heads given the
- two-sided coin times the probability of the
- two-sided coin.
- Well, this is the probability of 5 out of 5 heads given
- the two-sided coin, you're guaranteed to get 5 out of 5.
- And the probability of getting a two-sided
- coin is 1 out of 10.
- This is this.
- So this is equal to 1/10.
- So my probability of just putting my hand in the bag,
- taking out a random coin and then flipping it five times and
- getting all 5 heads, is going to be equal to this plus this.
- So it's 9/320.
- plus 1/10.
- But let's make that so it's 320.
- That's 32/320.
- So 9 plus 32 is 41/320.
- So that's interesting.
- But now let me ask you even a more interesting question.
- This is why I think what we've learned so far is useful.
- Let's say that I do get 5 out of 5 heads.
- So I know that I got 5 out of 5 heads.
- So let me clear this up a little bit just because I think
- it's getting over-- So let's say that I do get
- 5 out of 5 heads.
- What is the probability that I picked the two-sided coin?
- So what we're asking is, what is the probability that I
- picked the two-sided coin, given that I got 5
- out of 5 heads?
- And I hope you understand what I mean here.
- Before we said, oh, for example, I hope you see the
- difference between this and what is the probability of
- getting 5 out of 5 heads, given the two-sided coin.
- If I know I have a two-sided coin, I know that the
- probability is equal to 1.
- I know that I'm guaranteed to get 5 heads.
- But this is a little bit more interesting.
- If I flip a coin five times in a row and I get a bunch of
- heads, I'll say, well, I either got maybe really lucky with a
- normal coin or I picked the two-sided coin.
- And I don't know which one happened, so
- there's a probability.
- And of course, the more times you get a heads, just
- intuitively, if I got 100 heads in a row I'll say, well, I
- probably picked the two-sided coin without actually just
- looking at the coin.
- We're assuming that that's not acceptable.
- And if I got a million heads in a row we'd be really, really
- sure that we picked the two-sided coin, but we're
- not completely sure.
- There's some probability that I picked the fair coin and it
- just happened to get heads a million times in a row.
- So how do we think about this?
- Well, I'll do it analytically and I'll draw it out, and
- hopefully, both of them will make sense to you.
- And I'll probably have to do it in the next video.
- I'll see you soon.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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