Probability (part 2) Let's flip a coin.
Probability (part 2)
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- Welcome back.
- Whenever I start running out of time I rush things and I
- most probably confuse you.
- So let me go over that last example one more time.
- So I said I have a completely fair coin and I'm going to
- flip it twice, and I want to know the probability that
- I get heads in both times.
- Well, we already know the probability that I get heads
- the first time is 1/2.
- There's 1/2 chance I get heads the first time and there's
- 1/2 chance that I get tails the first time.
- And then I'm going to flip again, and let's say, in this
- world, there's a 1/2 chance that we enter into this reality
- where the first flip is heads, and then in that world I'm
- going to flip a coin again.
- And I know that there's a half a chance of heads-- this is
- the second flip-- and 1/2 chance of tails.
- And I know in this other world-- I'll just draw it
- so we can figure out maybe other probabilities.
- I know that there's 1/2 chance of heads, so this
- is for the second flip.
- And I know that there's 1/2 chance of tails
- on the second flip.
- And just so you know, this is called a probability tree.
- And in general, it works excellent if you're not
- dealing with huge numbers.
- If you're not dealing with-- I'm doing 100 trials or the
- probabilities get really messy or in each trial there's 50
- different circumstances that can occur.
- In this situation there's two branches per-- you could say
- per node, so there's not that many circumstances that
- can occur on each trial.
- And we're not doing that many trials, so it's
- pretty manageable.
- And in these situations, the probability tree
- works really well.
- You can never get overwhelmed with the information then.
- And I'll show you, we'll do some pretty complicated
- examples using probability tree.
- There's no reason why I actually circled that with
- yellow; I just happened to switch to yellow.
- But anyway, just so you know, this was the first trial.
- This is the second trial.
- We want to know the probability of getting here because we got
- heads on the first trial, heads in the second.
- Well, we know that there was a 1/2 chance of getting there and
- then we know of that 1/2 or you know, 1/2 the times we got
- here, 1/2 of those times you would get there.
- So the probability of getting there is 1/2 times 1/2,
- which is equal to 1/4.
- Another way of thinking about it is each of these scenarios--
- this is a scenario heads, heads because you got a heads
- and then a heads.
- This is the scenario heads, tails.
- This is the scenario tails, heads.
- You got a tail, then you got a heads.
- And this is the scenario tails, tails.
- And all of these are equally probable scenarios.
- There's no reason why one is more likely than the other.
- And so we have 4 equally probable scenarios, so
- the total of equally probable scenarios is 4.
- How many of these equal probable scenarios is our
- probability or our event true?
- Well, it's one of them-- heads heads.
- So it's 1/4.
- So that's another way of viewing it.
- Another way you could view it is if we did this trial a
- hundred times, if we did is a hundred times, 50 of the times
- my first flip will be heads.
- And then of those 50 times 25 of those times-- half of those
- trials where I got 50 the first time will be heads again.
- And so 25 out of the total 100 trials will end up with heads
- heads, or this is 25% chance.
- I wanted to show you that all of these are pretty much the
- equivalent, but I want your brain to kind of connect
- and make sure that it understands everything.
- So now that we've drawn this probability tree I don't want
- it to go to waste, so let me give you another circumstance.
- What's the probability I get one heads and one tail?
- And notice, I'm not saying that it has to be in that order.
- You know, it could be tails, heads or heads, tail.
- Let me write that down.
- This is equal to the probability of tails, heads
- plus the probability of heads, tails.
- Or we could write that as-- and this is kind of with set theory
- notation, but it's good to be familiar with all of these.
- It's a probability of getting tail, heads or this U looking
- thing means or in set theory.
- Or the probability of getting heads, tails, right?
- We can get tails, heads or heads, tails.
- In either of these we're getting one heads
- and one tails.
- So what's that probability?
- Well, we can just look at all of the outcomes and we know
- that there are 4 equally probable outcomes.
- And then, how many of these is the event that we want true?
- Well, it's 1, 2.
- So there's a 2/4 or equal 1/4 chance where when we do both of
- these flips we get at least one heads and at least one tails.
- And I think that's a good time now that we've drawn this tree
- and we've talked about two flips in a row, to realize that
- these are mutually exclusive events.
- The reason why we can multiply the probability of the first
- event times the probability of the second event is because the
- probability of getting heads the second time is completely
- independent of whether I got heads the first time.
- And that might be obvious to you, but sometimes it isn't.
- You know, some people feel that if they got 5 heads in a row
- that they're more likely to get a tails on the sixth time.
- But what we know from experience really, is that no,
- it's not like the coin knows that you got 5 heads in a row
- and then it says, oh boy, I better give Sal a tails now
- just to kind of make up for all the 5 heads he got.
- It's equally problems to get a heads again.
- So that's just something to keep in mind.
- And it's amazing how just human beings psychologically-- you
- feel that if you have a big streak of heads that
- you're due for a tails.
- But your probability of getting that tails does not increase.
- Although, if you keep doing it you're probability of
- eventually-- the more times you do it, the probability of
- getting a tails does increase and I'll show you that soon.
- That last statement probably confused you and I probably
- shouldn't have said it.
- Let's do an example.
- Now that we know that in mutually exclusive events
- you can just multiply the probabilities.
- So let's say I have a fair coin and what's the probability
- of getting 5 heads in a row?
- Well, I have a 1/2 chance of getting heads the first time.
- Then of those, 1/2.
- Then another 1/2.
- So it's 1/2 times 1/2 times 1/2 times 1/2 times 1/2.
- And that equals 1/2 to the fifth power.
- That's equal to 1 over 2 to the fifth power.
- And that's 1/32.
- That's the probability of getting 5 heads in a row.
- What's the probability of getting-- let's say I were
- to flip a coin seven times.
- So let's say I flip a coin, out of seven times, what's the
- probability of not getting any heads?
- I'm going to switch colors just for the hell of it.
- Well that might seem a little bit more convoluted to you,
- but just think about it.
- The probability of out of the seven flips of not getting
- any heads, that is equal-- what has to be true then?
- We got all tails then.
- That's equal to the probability of getting 7 tails in a row.
- And what does that equal?
- Well, the tails are equally likely to the heads.
- So that would be 1/2 to the seventh power.
- That equals-- what is that?
- So it'll be 4 times 132, so that's equal to 1/128.
- And a fun experiment to do is maybe in your classroom or if
- you have a large group of people together, maybe 128 or
- some large number of people and you make everyone flip
- a coin multiple times.
- By the end of 5 or 6 or 7 flips, there will probably be--
- and later on we'll actually figure out the probability
- that there is someone who has a streak.
- But there will probably be someone who has a very long
- streak of heads or tails.
- And that person will probably think that there's something
- magical about what is going on in their life at that moment.
- And it's important to realize if you have a large room, if
- you have a room of 500 people, someone's going to be getting--
- there's a likelihood and we haven't figured that out yet,
- but just given that you have a 1 in 128 chance of getting 7
- heads or 7 tails in a row.
- Someone's going to get if you have 500 people flipping coins.
- But that person, they just view their own reality, they think
- that today is their lucky day.
- And they think that they're on some type of streak and that
- they should go to Vegas.
- But that's just something that you should think about because
- probability can often play with our brains.
- But anyway, I have 10 seconds left, so I will
- see you in the next video.
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