Getting Exactly Two Heads (Combinatorics) A different way to think about the probability of getting 2 heads in 4 flips
Getting Exactly Two Heads (Combinatorics)
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- Im going to start with a fair coin and I'm going to flip it four times
- and the first question i ask is what is the exact probability
- exactly one head or heads which is one of those confusing things
- when you're talking about what side of the coin
- and I know I have been not doing this consistently
- I'm tempted to say if you're saying one it feels like
- you should do the singular which would be head
- but I read it up on the, I read up a little bit of it
- on the internet and it seems like when you're talking about coins,
- you really should say one heads which seems a little difficult for me
- but I'll try to go with that
- So what is the probability of getting exactly 1 heads?
- And I put that in quots to see the point, which is really...
- We are just talking about 1 heads there.
- But it's called "heads" when you are doing with coins.
- Anyway I think you get what I am talking about.
- Let's think about how many different possible ways
- we can get four flips over a coin.
- So we are going to have one flip, then another flip,
- then another flip, then another flip.
- And this first flip has two possibilities that could be heads or tails
- The second flip has two possibilities that could be heads or tails.
- The third flip has two possibilities could be heads or tails.
- And the fourth flip has two possibilities that could be heads or tails
- So you have 2 times 2 times 2 times 2
- which is equal to 16 possibilities.
- 16 possible outcomes when you flip a coin 4 times.16 possible outcomes
- And any one of the possible outcomes would be 1/16.
- So if I want to say the probability
- I am gonna not talk about this one head. If I just take,
- just you know maybe this thing has three heads right here.
- This is exact sequence of the event.
- This is the first flip, second flip, third flip, fourth flip,
- getting exactly this. This is exactly 1 out of possible of 16 events.
- Now with that out of the way, let's think about
- how many of those 16 possibilities involved getting exactly one heads?
- We can list them. You could get your heads. So this is equal to
- the probability of getting the heads in the first flip,
- plus the probability of getting the heads in the second flip,
- plus the probablity of getting the heads in the third flip,
- remember exactly one heads, nothing at least 1, exactly 1 heads.
- So the probability in the third flip,
- and then or the possibility you get the heads in the fourth flip,
- tails heads and tails.
- And we know already what the probability of each of these things are.
- There are 16 possible events.
- Each of these are one of those of 16 possible events.
- So this is going to be 1/16, 1/16, 1/16 and 1/16.
- And so we really saying the probablity of getting exacly one heads
- is the same thing as the probability of getting heads
- in the first flip or heads in the second flip,
- or heads in the third flip or heads in the fourth flip.
- And we can add the probablities of these different things
- because they are mutually exclusive.
- Any two of these things can not happen in the same time.
- You have to pick one of these scenarios.
- So we can add the probabilities 1/16 plus 1/16 plus 1/16
- plus 1/16 that gets 4 times of all.
- Assume that I did. So you'll get 4/16 which is equal to 1/4.
- Fair enough. Now let's ask a slightly more interesting question.
- Let's ask ourselves, the probability of getting exactly 2 "heads".
- There is a couple of ways we can think about it.
- One is just in the traditional way.
- Let's just, the way you know, the number of possibilities,
- equally likely possibilities
- You could only use methodology because it is a fair coin.
- So how many of the total possibilities have the two heads
- of the total of the equally likely possibilities.
- So we know there are 16 equally likely possibilities.
- How many of those have two "heads"?
- So I actually ahead of times so we save time.
- I have drawn all of the 16 equally likely possibilities.
- And how many of these involed two heads?
- Well let's see this one over here has two heads.
- This one over here has two heads. This one over here has two heads.
- Let's see that's this one over here has two heads.
- And this one over here has two heads.
- And then this one over here has two heads.
- And I believe we are done after that. So if we count them 1,2,3,4,5,6
- of the possibilities have exactly two heads.
- So 6/16 equally likely possibilities have two heads.
- So we have (what is this) 3/8 chance of getting exactly 2 heads.
- Now that's kind of what we have been doing in the past.
- But I do think about a way
- so we wouldn't have to write out all the possibilities.
- And the reason why that is useful is we are going to do
- the four flips now and if we are doing with 10 flips,
- there is no way that we can write out all the possibilities like this
- so I really want a different way of thinking about it.
- And the different way of thinking about it is
- of course we are seeing exactly two heads.
- You can imagine what we're having the four flips.
- Flip 1, flip 2, flip3, flip 4, so these are the flips.
- You can see the outcomes of the flips.
- And you are going to have exactly 2 heads. You can sayŁŹ
- "Well look, I am going to have 1 head in one of these positions
- and then 1 head in the other positions."
- So how many... If I am picking the first, so you know,
- you could say I have kind of Heads 1 and I have a Heads 2.
- And I don't want you to think that these are
- somehow the heads in the first flip or the heads in the second flip.
- What I am saying is we need 2 heads.
- We need in total of 2 heads in all of our flips.
- And I am just giving one of the heads a name
- and I am giving the other heads a name.
- And we are going to see in a few seconds
- that we actually don't want a double count.
- We don't want to double count the situation: Heads 1, Heads 2,
- tails, tails and Heads 2 Heads 1, tails tails.
- For our propers, these are exact same outcomes.
- So we don't want to double count that.
- We are going to have a count for that.
- But if we just think about it generally,
- how many different flips can that first heads show up in?
- Well this 4 different flips that first heads could show up in.
- So there are 4 possibilities, 4 flips or 4 places that can show up in.
- Well that first head takes up one of these 4 places.
- Let's just say the first heads shows up on the third flip.
- Then how many different palces can that second head show up in?
- Well if that first head is in one of 4 places
- and then that second head can only be in three different places,
- so that second head can only be (let me put it in nice color here)
- can only be in three different places.
- And so you know could be any one of these right over there,
- any one of those three places.
- And so when you think about it in terms of the first,
- I don't want to say the first head, Head 1,
- actually let me call Head A and Head B,
- that what you want to me to talk about it,
- first flip or the second flip.
- So this is Head A and this right over there is Head B.
- So if you had a particular, I mean these heads are identical.
- These outcomes are different.
- But the way we talk about right now, it looks like 4 places that
- we can get this Head A and there 3 places we can get this Head.
- So if you multiply all of these different ways,
- you could get all of the different scenarios.
- This is 4 diferent places and this is in one of the 3 left places.
- You get 12 different scenarios. But there are only be 12 scenarios
- if you have used this as being different than this.
- Let me rewrite it with our news. This is Head A. This is Head B.
- This is Head B. This is Head A.
- There will be only 12 different scenarios
- if you use these two things as fundementally different.
- But we don't. We actually double counting because
- we can always swap these two heads and have the exact same outcome.
- So you want to do is actually divided by 2.
- So you wanted to divided by all of different ways
- that you can swamp 2 different things.
- If we had 3 heads here you will think about all the different ways
- that you could swamp 3 different things
- if we have 4 heads here be all of the different ways,
- you could swamp 4 different things.
- So you have there 12 different scenatios
- if you couldn't swap them but you want to be divided
- by all of the different ways you could swap 2 things.
- So 12 divided by 2 is equal to 6, 6 different scenarios,
- fundementally different scenarios considering
- that you could swamp them.
- If you assume that Head A and Head B can be interchangeble
- that it's completely identical outcome for us
- because there are really just heads.
- So there are 6 diffrent scenarios
- and we know that there is a total of 16 equally likely scenarios.
- So we could say the probability of getting exactly 2 heads is 6 times,
- 6 scenarios, and we can do this a couple ways.
- There are 6 scenarios gives us 2 heads of possible 16
- or you could say there are 6 possible scenarios
- and the probability of each of the those scenarios is 1/16
- but either way you will get the same answer.
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