Probability using Combinations Probability of getting exactly 3 heads in 8 flips of a fair coin.
Probability using Combinations
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- So you might be wondering why I went off into permutations and
- combinations in the probability playlist, and I think you'll
- learn in this video.
- So let's say I want to figure out the probability-- I'm going
- to flip a coin eight times and it's a fair coin.
- And I want to figure out the probability of getting
- exactly 3 out of 8 heads.
- So I say 3/8 heads, but 3 of my flips are going to be heads and
- the rest are going to be tails.
- So how do I think about that?
- Well, let's go back to one of the early definitions
- we used for probability.
- And that says, the probability of anything happening is the
- probability of the number of equally probable events into
- which what we're stating is true.
- So in which the number of events-- I guess trials or
- situations-- in which we get 3 heads, and exactly 3
- heads, we're not saying greater than 3 heads.
- So 4 heads won't count and 2 heads won't count, 5 heads
- won't-- only 3 heads.
- And then, over the total number of equally probable trials--
- not trials, total number of equally possible outcomes.
- I should be using the word outcomes.
- So just with the word outcomes it should be the total number
- of outcomes in which what we're saying happens.
- So we get 3 heads over the total possible outcomes.
- So let's do the bottom part first.
- What are the total possible outcomes if I'm flipping
- a fair coin eight times?
- Well, the first time I flip it I either get heads or
- tails, so I get 2 outcomes.
- And then when I flip it again I get 2 more come
- for the second one.
- And then, how many total outcomes?
- Well, that's 2 times 2 because I could have got 2 in the
- first, 2 in the second flip.
- And then essentially we would multiply 2 times
- the number of flips.
- So that's 5, 6, 7, 8, and that equals 2 to the eighth.
- So the number of outcomes is just going to be 2 to the
- total number of flips.
- And hopefully that make sense to you.
- If not, you might want to re-watch some of
- the earlier videos.
- But that's the easy part.
- So there's 2 to the eighth possible outcomes when you
- flip a fair coin eight times.
- So how many of those outcomes are going to result
- in exactly 3 heads?
- Let's think of it this way.
- Let's give a name to each of our flips.
- Let's give a name to them.
- So let me make a little column, we'll call these the flips.
- This is my flips column.
- And I could name them anything.
- I could name them Larry, Curly, Moe.
- I could name them-- well, I would need 5 more names for
- them, but I could name them the 7 dwarfs or the 8 dwarfs really
- because I have 8 flips.
- I'll number the flips.
- Flip 1, 2, 3, 4, 5, 6, 7, 8.
- And I'm the god of probability.
- And essentially, I need to just pick 3 of these flips that
- are going to result in heads.
- So another way to think about it is, these could be 8 people
- and I could pick which of these-- how many ways can I
- pick 3 of these people to put into the car?
- How many ways can I pick 3 of these people to sit in chairs.
- And it doesn't matter the order that I pick them in.
- It doesn't matter if I say the people that are going to get in
- the car are going to be people 1, 2, 3 3.
- Or if I say 3, 2, and 1, or if I say 2, 3, and 1.
- Those are all the same combination.
- So similarly, if I'm just picking flips and I have
- to say, OK, 3 of these flips are going to get
- into the heads car.
- Heads is like they're sitting, they're people sitting down.
- I don't want to confuse you too much.
- But essentially I'm just going to choose 3
- things out of the 8.
- So I'm essentially just saying, how many combinations can I get
- where I pick 3 out of these 8.
- And so that should immediately ring a bell that we're
- essentially saying, out of 8 things we're going to choose 3.
- How many combinations of 3 can we pick of 8 and that we
- went over in the last video.
- And let's do it with the formula first.
- So let me write the formula up here just so you remember it,
- but I also want to give you the intuition again,
- for the formula.
- So in general, we said, n choose k, that is equal to n
- factorial over k factorial times n minus k factorial.
- So in this situation that would equal 8 factorial over
- 3 factorial times what?
- 8 minus k-- times 5 factorial.
- Or another way of writing this, this would be 8 times 7 times 6
- times 5 times 4 times 3 times 2 times 1 over-- I'll just write
- 3 factorial for a second.
- Then times 5 times 4 times 3 times 2 times 1.
- And of course, that and that cancel out and all you're
- left with is 8 times 7 time 6 over 3 factorial.
- And I did this for reason because I want you to re-get
- the intuition at least for this part of the formula.
- That's essentially just saying, how many permutations can I--
- how many ways can I pick 3 things out of 8?
- And that's essentially saying, well, before I pick anything
- I could pick 1 of 8.
- Then I have 7 left to pick from for the second spot.
- And then I have 6 left to pick for the third spot.
- And so that's essentially the number of permutations.
- But since we don't care what order we picked them in, we
- need to divide by the number of ways we can rearrange 3 things,
- and that's where the 3 factorial comes from.
- And so hopefully I didn't confuse you, but if I did you
- can go back to this formula for the binomial coefficient.
- But it's good to have the intuition.
- And then once we're at this point we can
- just calculate this.
- Well, what's this?
- This is 8 times 7 times 6 over 3 factorial of
- 3 times 2 times 1.
- So that's 6.
- The 6 cancels out, so it's 8 times 7.
- So there's 8 times 7, or what is that?
- That's equal to 56.
- So there's 56 different ways to pick 3 things out of 8.
- Or if I have 8 people there's 56 ways of picking 3 people to
- sit in the car or however you want t view it.
- But if I have 8 flips there's 56 ways of picking 3 of
- those flips to be heads.
- So let's go to our original probability problem.
- What is the probably that I get 3 out of 8 heads?
- Well, it's the number of ways I can pick 3 out of those 8,
- so it equals 56, over the total number of outcomes.
- The total number of outcomes is 2 to the eighth.
- Another way I could write that-- 56, let me unseparate.
- That's 8 times 7 over 2 to the eighth.
- 8 is 2 to the third.
- Let me erase some of this.
- Not with that color.
- Let me erase that.
- Let me erase all of this just so I space.
- And I will switch colors for variety.
- Let me use the small pen.
- OK, so I'm back.
- All right, so 8 is the same thing as 2 to the third times
- 7-- this is all just mathematical simplification,
- but it's useful-- over 2 to the eighth.
- And so, if we just divide both sides-- the numerator and the
- denominator by 2 to the third, this becomes 1.
- This becomes 2 to the fifth.
- And so it becomes 7/32.
- Is that right?
- So if I were to pick 3 out of 8-- yep, I think that is right.
- And so what does that turn out to be?
- Let me get my calculator.
- to make careless mistakes.
- Let's see.
- My calculator seems to have disappeared.
- Let me get it back.
- There it is.
- 7 divided by 32 is equal to 0.21875.
- Which is equal to 21.-- you know, if I were to round
- roughly-- 21.9% chance.
- So there's a little bit better than 1 in 5 chance that I get
- exactly 3 out of the 8 flips as heads.
- Hopefully I didn't confuse you and now you can apply that
- to pretty much anything.
- You could say, well, what is the probability of getting-- if
- I flip a fair coin-- of getting exactly 7 out of 8 heads?
- Or you could say, what's the probability of getting
- 2 out of 100 heads?
- And you could use it the exact same way we did this problem.
- I'll see you in the next video.
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