Old school probability (very optional)
Probability (part 2) Let's flip a coin.
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- 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.
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?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.