Probability (part 7) More on conditional probability. Touch on Bayes' Theorem.
Probability (part 7)
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- So let's think about what we did in that last video and
- maybe do it in a little bit more of a visual way.
- So let's say this rectangle that I'm about to draw is
- the set of all the things that I guess can happen.
- It's a set of all of the outcomes.
- And we know from the get-go, and we're still doing that same
- example where I have a bag of coins and one of them
- only has heads on it.
- Two sides are heads, the other ones are normal.
- So, if this is the whole set of things we know that there's a
- 1/10 chance that we pick the coin that only has heads.
- So let me say out of this area-- I don't
- know-- this is 1/2.
- That's 1/4.
- So 1/10 is going to be about that much, roughly.
- So this area right here.
- And let me color it.
- Let me color it a color.
- So that area right there, this is the chances that we got
- the two-sided sided coin.
- And this whole area, this is all a normal coin.
- And then we said, oh, you know, what are the chances that
- we get 5 heads in a row?
- Well, if we had a normal coin, there was a 1 out of 32 chance
- that we get 5 heads in a row.
- So whatever this area is, 1 out of 32.
- So I'll do it in a column.
- It's roughly there.
- It can be a little messy.
- And I'm assuming that this column right here is 1/32
- of this normal area.
- So this is a situation of normal-- I'll just write this
- as n-- and that upside down U means and in set theory.
- Or it's a kind of intersection.
- But it means and I got 5 out of 5 heads.
- And if we think about, what is the area of this?
- Well, it's going to be the area-- it's 1/32
- of the normal area.
- And the normal area is 9/10.
- So this is equal to the normal area, which is 9/10.
- And then 1/32 of that.
- Times 1/32.
- Or you could also, another way of viewing that area, you could
- say that well, that area, that's the probability-- I'll
- do it in the same color.
- That's the probability of me getting-- I'll just
- write 5 out of 5.
- 5 out of 5 means 5 out of 5 heads.
- That's what probability of getting 5 out of 5 heads
- given that I picked a normal coin out.
- Times the probability that I picked out a normal coin.
- And so this is 1 out of 32, and this is 9 out of 10.
- And I'm assuming that the area of this entire magenta
- rectangle I already drew, that's 1 or 100%.
- Or however you want to view it.
- And so this is 1 out of 32 times 9 out of 10.
- So this is 9 out of 32.
- So this whole thing, the area this entire of this entire
- magenta rectangle, is 1.
- Then the area of this that I just shaded in, this is 9/320.
- Another way of viewing this, this is also the probability--
- this area right here is also the probability that I got
- 5 out of 5 and that I picked the normal coin.
- So that's also equal to.
- Both of those things are true in this area.
- In this area, if I have a point in that area-- this might be a
- little abstract for you-- if at any point, it is true that I
- got 5 out of 5, and I have picked a normal coin.
- So that's also going to be equal to and I'm doing an
- arbitrary color change here.
- 9 out of 320.
- So we could say in general, that the probability of me
- getting 5 out of 5 coins, given that I got a normal coin times
- the probability that I got the normal coin-- that is going
- to be equal that 5 out of 5 happens and I got the normal.
- That equals probability of 5 out of 5, and I
- got the normal coin.
- Fair enough.
- Let's go to the other side of this.
- This kind of yellowish color here.
- There's a probability I got a two-sided coin.
- And then given that, what's the probability that
- I get 5 out of 5?
- Well, pretty much, this entire area.
- It's also true that I get 5 out of 5.
- So the probability that I get 5 out of 5 and two-sided, it's
- equal to the probability-- oh sorry, and.
- And two-sided, is equal to the probability of 5 out of 5,
- given the two-sided coin times the probability
- of the two-sided coin.
- So this is 1, right?
- If I got the two-sided coin I'm definitely going to 5 out of 5,
- That's why this whole area is filled in.
- And then, the probability of the two-sided coin we know
- already is 1 out of 10.
- So this area is 1 out of 10.
- And so what is the total area where I have 5 out of 5?
- Well, it's going to be this area over here,
- which I figured out.
- Which was 9/320 of this entire rectangle.
- And then this whole area is also another area where I
- got 5 out of 5 because I picked the two-sided coin.
- And that's 1/10.
- In the last video we would essentially-- if we want to
- know the chances of getting 5 out of 5, it's this
- total rectangle.
- I say this without showing you anything, so let
- me-- it'll be red.
- And I think you get the point.
- That is the 5 out of 5, so it's going to be this
- area plus this area.
- And that is why it was 9 out of 320 plus 1 out of 10.
- And we had the answer last time.
- Fair enough.
- So let me pose a slightly more difficult problem.
- I don't know if I should have just erased that, but I did.
- So let's move forward.
- What is the probability that I picked the two-sided coin given
- that I got 5 out of 5 heads?
- And remember, this is different than the probability that I get
- 5 out of 5 heads given the two-sided coin.
- What's the difference?
- This says I picked the two-sided coin, now what's the
- probability I got 5 out of 5?
- Well that's equal to 1.
- We're guaranteed to get a million heads in a row.
- But this says I put my hand in the bag, I didn't look at which
- coin I got, I just flipped it five times and I know that
- I got 5 heads in a row.
- So now I say, well, what is the probability without looking at
- the coin, what is the probability that this was
- the two-sided coin that I picked up?
- Well, we can go back to what we said before.
- Let me just say it a little bit abstract.
- Before I clear this, we know that the probability of a
- given b, or the probability of-- well, actually.
- Let me start over.
- The probability of a and b happening, if you remember that
- thing that before I erased it, is a probability of a given
- that b happened times the probability that b happens.
- If both of these things have to happen then we could say, well,
- definitely b has to happen.
- And then, whatever the probability of a happening
- given that b happened.
- This could be exclusive events.
- If these are independent this could be equal to the
- probability of a times the probability of b.
- But we don't know for sure that they're exclusive events.
- So the probability of a given b could be the probability of a.
- It might not be dependent on b at all, but we don't know that.
- Let's ignore that.
- So the probability of a given b is that.
- But we also know that the probability of a given
- b is equal to-- I mean, the probability-- sorry.
- We also know the probability of a and b is also the same thing
- as the probability of b and a happening.
- The same thing.
- And so that's the probability of b given that a happened,
- times the probability that a happened.
- By the same logic.
- Well, we see that this is equal to this, so this
- has to be equal to that.
- So let's write that down.
- And we're going to do a lot of applications; I know this is
- a little abstract right now.
- So that tells us the probability of a given b times
- the probability of b, is equal to the probability of b given
- a, times the probability of a.
- And then, let's just divide both sides by a.
- And we get this, the probability of b given
- a is equal to this, probability of a given b.
- Times the probability of b divided by the
- probability of a.
- And this is called Bayes' theorem or Bayes' law.
- And this is what we will now use to figure out the
- probability that we got the two-sided coin after seeing
- that we rolled or we got 5 heads in a row.
- See you in the next video.
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