Exactly Three Heads in Five Flips Probability of exactly 3 heads in 5 flips using combinations
Exactly Three Heads in Five Flips
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- So let's start again with a fair coin, and this time,
- instead of flipping it four times
- let's flip it five times
- Five flips of this fair coin
- And what I want to think about in this video
- is the probability of getting exactly 3 heads.
- And the way I'm going to think about it is
- if you have five flips how many different,
- equally likely possibilities are there?
- So you're going to have the first flip,
- and there's two possibilities there,
- heads or tails; second flip, two possibilities there;
- third flip, two possibilities; fourth flip,
- two possibilities; fifth flip, two possibilities.
- That's two times two times
- two times two times two (I hope I said that five times!).
- So, two, or you could use just two to the fifth power
- and that is going to be equal to thirty-two
- equally likely possibilities.
- Thirty-two: Two times two is four,
- sixteen times two is thirty-two possibilities.
- And to figure out this probability,
- we really just have to figure out
- how many of those possibilities involve getting three heads.
- We could draw it right out,
- all of the thirty-two possibilities,
- and literally just count the heads,
- but let's just use that other technique that
- we just started to explore in that last video.
- We have five flips here (Let me draw the flips).
- And we want to have exactly three heads,
- and I'm going to call those three heads
- Heads A, Heads B, Heads C, just to give them a name,
- although what we're going to see later in this video
- is that we don't want to differentiate between them.
- To us, it makes no difference if we get this ordering:
- Heads A, Heads B, Heads C, tails, tails;
- or if we get this ordering:
- Heads A, Heads C, or Heads B, tails tails.
- We can't count these as two different orderings,
- we can only count this as one.
- So what we're going to do is first come up with
- all of the different ordering
- if we cared about the difference between A and B and C.
- And then we are going to divide by the,
- all of the different ways
- that you could arrange three different things.
- So how many ways can we put A,B and C into these five buckets
- that we can view as the flips if we cared about A, B and C.
- So let's start with A.
- If we haven't allocate any of these buckets to
- any of the heads that we could say A
- could be in five different buckets,
- so there is five different possibilities where A could be.
- So let's just say, that you know,
- this is the one that
- goes in although could be any one of these five.
- But this takes one of the five,
- then how many different possibilities can these heads sit in?
- How many different possibilities are there?
- Well then there is only going to be four buckets left,
- so then there is only four possibilities.
- And so if this was where the Heads A goes
- and Heads B can be any of the other four.
- Heads A was in this first one and then Heads B
- could be in any of the other four.
- I'll do a particular example maybe Heads B,
- shows up right there.
- So once you take two of the slots,
- how many spaces do we have for Heads C?
- Well we only have three spaces left then, for Heads C.
- That can be any of these three spaces that could just
- show particular example of what looked like that.
- If you cared about order,
- hwo many different ways can you out of five different spaces,
- allocated them to three different heads?
- You would say it is five times 4 times 3.
- Five times four is twenty times three is equal to sixty.
- So you will say there are sixty different ways to
- arrange Heads A, B and C in five buckets or five flips
- or these are people in five chairs.
- And obviously there are sixty possibilities of
- getting three heads
- if I had only 32 equally likely possibilities.
- And the reason why we got such a big number over here is that
- we are counting this scenario as being fundamentally
- different than this was Heads B,
- Heads A and then Heads C, over here.
- And what we need to do is to say
- they aren't different possibilities.
- We don't have to, kind of,
- we don't have to overcount for
- all of the different ways you arrange this.
- And so what we need to do is divide this
- by all of the different ways that
- you can arrange three things.
- So if I have three things that in three spaces,
- so here I have a Heads in the second flip,
- third flip and fifth flip.
- If I have three things in three spaces like this,
- how many ways can I arrange them?
- So if I have three spaces, how many ways can I arrange?
- And A, B and C in those three spaces.
- Well A can go into three spaces.
- It can go any of those three.
- A can go into any of the three spaces.
- Then B would have two spaces left once A takes one of them.
- And then C would have one of these spaces left,
- once A and B take two of them.
- So there is three times two times one way
- to arrange three different things.
- So three times two times one is equal to six.
- So the number of possibilities of getting three heads
- Let me write this down in another color.
- So the number of possibilities is equal to this
- five times four times three over the number of ways that
- I can well arrange three things.
- We don't want to overcount for all of these,
- doing this arrange is fundamentally different
- than this arrangement.
- Then we want to divided by three times two,
- I want to do this in same orange color,
- dividing by three times two times one.
- Three times two times one and which gives us,
- the numerator is one hundred and twenty,
- one hundred and twenty divided by six, sorry,
- it's sixty divided by six, which gives us ten possibilities.
- This gives us exactly three heads.
- And that's of 32 equally likely possibilities.
- So the probability of getting eaxctly three heads,
- when you getting exactly three heads
- in 10 of the 32 equally likely possibilities.
- So you have a 5/16 chance of that happening.
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