Probability using combinatorics
-
Example: Probability through counting outcomes
-
Example: All the ways you can flip a coin
-
Getting Exactly Two Heads (Combinatorics)
-
Probability and Combinations (part 2)
-
Probability using Combinations
-
Exactly Three Heads in Five Flips
-
Example: Different ways to pick officers
-
Example: Combinatorics and probability
-
Example: Lottery probability
-
Mega Millions Jackpot Probability
-
Generalizing with Binomial Coefficients (bit advanced)
-
Conditional Probability and Combinations
-
Birthday Probability Problem
-
Probability with permutations and combinations
Exactly Three Heads in Five Flips Probability of exactly 3 heads in 5 flips using combinations
⇐ 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.
- 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.
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.