Old school probability (very optional)
Probability (part 8) Introduction to Bayes' Theorem
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- So in the last video I hopefully proved to you, at
- least maybe it doesn't make intuitive sense just yet, but I
- showed you that the probability of a and b is equal to the
- probability of a given b times the probability of b.
- And the probability of b given a is equal to the probability
- of b given a times a.
- And then we played around a little bit and we got this.
- And this is called Bayes' theorem or Bayes' law.
- And I will now show you how you can apply this formula.
- But it shouldn't be in a formula.
- You should always be able to intuitively come to this
- conclusion, even if you forget Bayes' theorem, which I do all
- the time and then I go through that thought process that
- we did at the end of the last video over again.
- But let's see how we can apply this to this problem.
- So we want to figure out the probability that we pick the
- two-sided coin given that we observed when we flip it
- five times we got 5 heads.
- So let's say that this is event a and this is event b.
- So we can use Bayes' theorem.
- So the probability that I got the two-sided coin, given 5 out
- of 5 heads is equal to the probability-- and I'm actually
- going to think about what this fraction means in a second.
- But it means the probability that I got a given
- b, so it's this.
- The probability that I got 5 out of 5 heads, given the
- two-sided coin times the probability of the two-sided
- coin-- b-- we defined event b as the two-sided coin.
- Time the probability of two-sided coin.
- Divided by the probability of in general, the probability
- of getting 5 out of 5 heads.
- Well, what's the probability of getting 5 out of 5 heads
- given the two-sided coin?
- Well, we wrote that down.
- We're guaranteed to get 5 out of 5 heads, so that's 1.
- And what's the probability of getting the two-sided coin?
- Well, that was 1/10.
- And now what's the probability of getting 5 out of
- 5 heads in general.
- Well, we figure that out, I think, either in the last video
- or the video before that.
- I forgot the exact number.
- Actually, it was something like 1/10 plus 9/320.
- Right, so this is 32.
- It was 41/320.
- We figured that out in the previous problem.
- This is over 41 over 320.
- And so what did we get?
- We get this is equal to 1/10 over 41/320.
- Now what is that equal to?
- That equals to 1/10 times 320/41.
- Let's divide the top and the bottom by 10.
- So it equals 32/41.
- That's interesting.
- I actually find that really interesting.
- I don't know which coin I picked out of the bag, but I
- know that if I flip that coin five times and all 5 are heads,
- there's a 32 out of 41 chance-- this is maybe a little bit
- better than a 75% chance that I picked the two-headed
- coin we could say.
- And let's think about this visually a little bit.
- So if this is the entire universe of what could happen,
- and we said, 1/10 of that-- this is the two-sided coin,
- this is the fair coin.
- And then, out of all of that, the situation in which we get 5
- out of 5, well, those could be-- if we got the two-sided
- coin all of that is going to be 5 out of 5.
- And then some small fraction, 1/32 of the fare coin.
- So this yellow circle represents getting
- 5 heads in a row.
- So if I want to know what's the probability of the two-sided
- coin, all Bayes' theorem is saying, well, OK, we observed
- that we got the two-sided coin.
- Sorry, we observed that we got 5 out of 5 heads.
- So we now know that we're in this yellow universe because
- the yellow universe represents the area or all of the outcomes
- where we got 5 out of 5 heads.
- And then we say, well, out of those outcomes, what percentage
- of them involved me getting the two-sided coin?
- Well, if we look here, all of these-- sorry.
- Edit.
- If we look at it here, this is the area out of the yellow
- where I had the two-sided coin and I got the 5 out of 5.
- And so this is going to be roughly 32 out of 41
- of this yellow area.
- So that's a visual representation of
- Bayes' theorem.
- This says the probability of getting a two-sided coin,
- given I got 5 out of 5, that says-- well, OK.
- Let's look at the universe of you got 5 out of 5, and then,
- what percentage of that universe involved you getting,
- essentially the two-sided coin?
- And that's that.
- And that's all Bayes' theorem essentially tells us.
- So let's see if we can apply that.
- Let's do a more interesting one.
- Actually, I'm going to to leave it there and then in the
- next video I will apply Bayes' theorem to more
- interesting problems.
- See you soon.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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