Dependent events
Monty Hall Problem Presentation and analysis of the famous Monty Hall Problem
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- Let's now tackle a classic thought experiment in probability
- called the Monty Hall problem.
- And it's called the Monty Hall problem because
- Monty Hall was the game show host
- in Let's Make a Deal where
- they would set up a situation very similar to
- the Monty Hall problem
- that we're about to say.
- So let's say that on the show, you're presented with
- three curtains.
- So you're the contestant
- this little character right over there
- You're presented with three curtains:
- curtain number one, curtain number two, and
- curtain number three.
- And you're told that behind one of these three curtains
- there is a fabulous prize.
- Something that you really want, a car, or a vacation
- or some type of large amount of cash.
- And then behind the other two,
- and we don't know which they are,
- there is something that you do not want,
- say a new pet goat, an ostrich,
- or something like that, or beachball,
- something that is not as good as the cash prize.
- And so your goal is to try to find
- the cash prize.
- And they say: "guess which one"
- or "which one you would like to select".
- And so let's say that you select
- door number one,
- or curtain number one,
- then the Monty Hall and Let's Make a Deal crew
- will make it a little more interesting.
- They won't just show you whether or not you won,
- They'll then show you one of the other two doors,
- or one of the other two curtains.
- And they'll show you one of the other two curtains that does not
- have the prize.
- And no matter which one you pick,
- there will always be at least
- one other curtain that does not have the prize.
- There might be two if you pick right,
- but there will always be
- at least one other curtain
- that does not have the prize.
- And then they will show it to you.
- And so let's say that they
- show you curtain number three,
- and that curtain number three has the goat.
- And then they will ask you,
- do you want to switch to curtain number two,
- and the question here is,
- does it make a difference?
- Are you better off holding fast,
- sticking to your guns, staying with your original guess,
- or are you better off switching
- to whatever curtain is left,
- or does it not matter,
- it's just random probability and
- it's not going to make a difference
- whether you switch or not.
- So that is the brain teaser.
- Start - pause the video now,
- I encourage you to think about it.
- In the next video,
- we will start to analyze the solution
- a little bit deeper.
- Whether it makes any difference at all.
- So now I've assumed that you've unpaused it,
- that you've thought deeply about it,
- perhaps you have an opinion about it,
- now let's work it through a little bit.
- And at any point,
- I encourage you to pause it,
- and kind of extrapolate beyond
- what I've already talked about.
- So let's think about the game show from
- the show's point of view.
- So the show knows
- where there is a goat and
- where there isn't a goat.
- So that's door number one,
- door number two, and door number three.
- So let's say that our prize is
- right over here.
- So our prize is the car.
- Our prize is the car and we have
- a goat over here, goat over here and
- over here we also have, we have two goats
- in this situation.
- So what are we going to do as a game show
- remember, the contestants don't know this
- we know this.
- So the contestant picks door number one
- right over here,
- then we can't lift door number two
- cause there's a car back there.
- We're going to lift door number three
- and we're going to expose
- we're going to expose
- this goat, in which case it probably
- would be good for the person to switch.
- If the person picks door number two,
- then, we as the game show,
- can show either door number one or
- door number three,
- and then it actually does not make sense
- for the person to switch.
- If they picked door number three,
- then we have to show door number one
- because we can't pick door number two
- and in that case, it actually makes a lot of sense
- for the person to switch.
- Now, with that out of the way,
- let's think of the probabilities
- given the two strategies.
- So if you don't switch,
- Don't switch
- Or another way to think about this strategy is
- you always stick to your guns
- you always stick to your first guess.
- Well, in that situation,
- what is your probability,
- what is your probability of winning.
- Well, there are three doors,
- the prize is equally likely to be
- behind any of them,
- so there are three possibilities.
- One has the outcome that you desire.
- The probability of winning will be one third
- if you don't switch.
- Likewise, your probability of losing
- well, there are two ways you can lose
- out of three possibilities,
- going to be two thirds.
- And these are the only possibilities,
- and these add up to one over here.
- So don't switch, one third chance of winning.
- Now let's think about the switching situation.
- So let's say always, when you always switch
- let's think about how this might play out.
- What is your probability of winning;
- and before we can think about that,
- think about how you would win
- if you always switched.
- So if you picked wrong the first time,
- if you picked wrong the first time,
- they're going to show you this
- so you should always switch.
- So if you picked door number one,
- they're going to show you door number three
- you should switch!
- If you picked wrong door number three,
- they're going to show you door number one
- you should switch!
- If you picked wrong and switch,
- you will always win!
- Let me write this down.
- And this insight actually came from
- one of the middle school students
- in the summer camp that
- Khan Academy's running,
- it's actually a fabulous way
- to think about this.
- So if you pick wrong,
- so if it's step one: initial pick is wrong
- Initial pick is wrong.
- So you pick one of the two
- wrong doors.
- And then step two, you always switch
- You always switch.
- You will land on the car,
- because if you pick one of the wrong doors,
- they have to show the other wrong door,
- and so if you switch you are going to end up
- on the right answer.
- So what is the probability of winning
- if you always switch?
- Well, it's going to be the probability
- that you initially picked wrong.
- Well, what's the probability that you
- initially picked wrong?
- Well, there are two out of three ways
- to initially pick wrong.
- So you actually have a two-thirds chance
- of winning.
- There's a two-thirds chance you
- are going to pick wrong,
- and are going to switch into the right one.
- Likewise, what's your probability of losing?
- What's your probability of losing, given
- that you're always going to switch?
- Well, the way that you would lose is
- you pick right,
- you pick correctly,
- in step two they're going to show an empty,
- they're going to show one of the two empty,
- or non-prized doors,
- and then in step three you are going to
- switch into the other empty,
- Switch into the other empty door.
- But either way, you are definitely going to switch,
- so the only way to lose if you're
- always going to switch
- is to pick right the first time.
- Well what's the probability of you
- picking right the first time?
- Well that is only one-third.
- So what you see here is sometimes
- counterintuitive, but hopefully
- this makes sense as to why it isn't.
- You have a one-third chance of winning
- if you stick to your guns,
- and a two-thirds chance of winning
- if you always switch.
- Another way to think about it is,
- when you first make your initial pick,
- there's a one-third chance that it's there,
- and there's a two-thirds chance
- that it's in one of the other two doors.
- And they're going to empty out
- one of them, so when you switch,
- you essentially are capturing that
- two-third probability.
- And we see that right there.
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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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