Binomial Distribution 4 Using Excel to visualize the basketball binomial distribution
Binomial Distribution 4
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- In the last video we talked about the situation where I'm
- taking 6 shots on a goal.
- We knew my probability of making any given shot, and what
- we wanted to do is figure out the probability distribution of
- me making k shots, in general.
- We defined our random variable, x, as the number
- of shots I make out of 6.
- So I realized in that last example and that example is
- right back here, that the math gets a little hairy.
- You're dealing with 0.7 then times 0.3 to the fifth
- and all of that.
- So it got a little complicated and what better thing to do if
- you have numbers like this, even better than the
- calculator than to use Excel.
- And you'll see that Excel is a fairly powerful tool for
- probability and statistics.
- And in general, for a lot of types of simulations.
- If you don't want to actually write serious code you can
- actually do fairly serious things with something as
- seemingly not so simple, or not so serious as Excel.
- But it's actually quite serious.
- So let's see.
- So let's say that I'm going to take-- let's define n.
- I'm going to say that that is, I'm going to take 6 shots.
- The probability of me making any shot I'll say is 0.3.
- That's what we did in the last video.
- The probability of a miss, let me actually write it a
- little bit more explicitly.
- Probability of a make is equal to 0.3 and the probability of a
- miss I'm going to write-- and instead of just writing 0.7
- there because we know it is, let's just it a little bit
- dependent on the first one.
- So write out the formula, we'll say that's equal to 1 minus--
- and then we'll select this cell right here.
- It's equal to 1 minus the probability that I make a
- shot, And notice it just calculated it for us, 0.7.
- Let me zoom in a little bit.
- There you go.
- That might be easier for you to read.
- Because I know it gets really small on YouTube.
- And this is neat because now if we wanted to see what happens
- when this becomes 0.2 that automatically
- calculates to 0.8.
- So that's neat.
- Let's just put it back to 0.3.
- And now let's do a bunch of rows so that we can calculate
- the probability that our random variable, x, could be 0
- shots or 1 shot or whatever.
- So let's say that k-- let me make a row called k.
- So that's the number of shots we need to make.
- So 0, 1, 2, 3, 4, 5, 6.
- You can't see the sixth one right now, let me just go
- up here and now you can see everything.
- And then we need to figure out the binomial coefficient.
- Well actually, let's do it first.
- Let's say the probability of making any one of those,
- any one way of getting no shots, or any one way of
- getting exactly 3 shots.
- So that's going to be the probability of a make.
- So you're going to make it.
- You're going to make k shots times the probability
- of a miss.
- So if you make k shots, how many do you miss?
- You miss n minus k.
- And I'm not doing anything fancy in Excel right here.
- I'm just kind of writing a label that hopefully is a
- little intuitive for you.
- So this is the probability-- this column is going to be-- in
- this cell right here is going to be the probability that any
- one particular way of making 3 shots and missing 3 shots.
- It's going to go in that cell eventually.
- And then you need to know that you needed the
- binomial coefficient.
- It's going to look a little bit messy now.
- So you want n choose k.
- So essentially this row is going to say, what's the
- probability for example, in this cell right here we'll have
- the probability of making the first and missing the next 5,
- or missing the first 5 and making the last 1.
- They're all going to be the same probability.
- You're going to have 1 make and 5 misses.
- And then, this cell we're going to say, well, how
- many different ways can I make just 1 out of 6?
- And that's why I'm going to choose 1 out of 6.
- k is 1 and n is 6.
- That's going to go there.
- And then we can calculate what the probability that our random
- variable is equal to k, is equal to this value.
- And it looks all fancy, but it's exactly what we did in
- the last video with my little doodle tool that I use.
- Which is actually just Microsoft Paint for those of
- you wondering because I get a lot of e-mails and I'm tired
- of saying it's just Microsoft Paint.
- All right, so what's the probability that I make 0
- and I miss 6 because that's what they're saying here.
- So this is going to be the probability of me making a
- shot-- and I'll put those dollar signs.
- That kind of fixes it on this cell.
- The probability that I make a shot to the kth power.
- So I make 0 shots times the probability that I miss a shot.
- Put those dollar signs there so that it fixes on that cell--
- to the n minus k power.
- In this case it would be the sixth power.
- I miss 6 six shots, so to the-- let me put it
- in parentheses-- n.
- Let me fix on that cell.
- n minus k power.
- And there you go there.
- That's the probability for example, of me making the
- first and missing the next 5.
- Or this could be the probability of me getting the
- first shot, then missing the second, and getting
- the last 4 missed.
- So any of one of those situations.
- And how many of those situations are there?
- Well that's n choose k.
- So that's equal to-- n choose k-- that's the
- same thing we did before.
- That's all this factorial stuff here.
- That's n choose k.
- So I'm just going to express the binomial coefficient
- expression in Excel.
- So it's a factorial of how many we're choosing from, how many
- shots we're taking, and the Excel function for
- that is fact.
- So we'll take the factorial of 6 and we'll divide it by--
- put a parentheses here.
- The factorial of k times the factorial of n.
- Let me fix that n.
- Minus k.
- And I know it fell off.
- n minus k.
- You put some parentheses there, so the parentheses match up.
- Actually, this what fixes on this cell Let me
- put an F4 there.
- And then there you go.
- There's only one way to choose 0 things out of 6 things.
- That's what that tells us.
- And then the probability that my random variable is equal to
- k, or in this case, that I make exactly 0 shots is equal to the
- probability of any of the specific ways of making 0 shots
- times the number of those ways there are times that.
- So that's the probability of making 0 shots.
- And this is a cool thing about Excel, we can now
- select these cells.
- And just go into this right hand corner, just drag it
- down, and it'll do that calculation for all of them.
- And this is neat because it calculated the
- binomial coefficients.
- This is 6 choose 3.
- n is 6, k is 3.
- This is 6 choose 3 is 20.
- It's nice and symmetric like you would expect.
- But the probabilities aren't that symmetric because
- we have 0.3 and 0.7.
- So it's not like the flipping a coin example that was
- 0.5 on either side.
- Sorry, these aren't symmetric and then when you multiply
- these, of course, these don't looks symmetric.
- But it's hard to look at these numbers, so actually, let's
- just use the powers of Excel to graph this probability
- So let's say insert chart.
- I'm always, I'm clumsy at inserting charts,
- so let me see.
- The data-- I think this is how I do it.
- I select the data.
- There you go.
- And then I need to select the label, the category x labels.
- OK, that's going to be that right there.
- And then let's see.
- Next, that looks good.
- I don't want to have anything.
- I don't need a legend.
- I don't want to show a legend.
- There you go.
- All right.
- Now I think I'm finished.
- Data labels-- yeah, that's fine.
- OK, finish.
- I don't want to get too fancy.
- Don't want the letters [INAUDIBLE].
- Let me make this bigger for you.
- So this is neat and I wish they didn't make the font so
- big, but you get the point.
- This is a discreet probability distribution of me making, in
- this case, k shots out of 6.
- And this was based on the fact that my probability of making
- any given shot was 0.3.
- Let me see if it'll actually-- and I'm not making any promises
- here-- let me see if I can actually make it so
- that it changes.
- If I make it 0.2 does it change?
- Oh, it changed!
- Look at that.
- That was neat.
- Let me bring this up a little bit.
- So what's neat is if I have a 30% chance of making a free
- throw, that is my probability distribution, my discrete
- probability distribution of making k shots.
- This is my probability of making no shots.
- I think it was like roughly-- oh, it even gives us it.
- Tell us it's 0.12% or it's 12% chance of making no shots.
- My probability of making 4 shots is only 6%.
- I know you can't see it but that little window that
- shows up right under my pointer says 0.06.
- So this is neat.
- And since the way we set it up we can actually change it,
- so that if I have a higher probability of making a shot,
- now it looks pretty symmetric.
- Now my chances of making 4 shots is the same as my
- chances of making 2 shots.
- And if I'm really good at making baskets, I have
- an 80% probability of making any given shot.
- Now, all of a sudden, the whole distribution has
- shifted to the right.
- But I just wanted to show you one, this is, I think, a
- pretty neat way to use Excel.
- And hopefully, it's not too daunting.
- The other thing is, these are all examples of binomial
- distributions and I wanted to give you an example that might
- be a little bit more relevant to your everyday life.
- The probabilities of making and missing aren't the
- same, they change.
- If they are the same then you end up with a
- situation like that.
- Let's say you're a horrible basketball player and you
- only have a 10% chance of making any given shot.
- So there's a very high probability that
- you make no shots.
- There's a 53% chance you make no shots and then there's a
- 35% chance you make one shot, and so forth and so on.
- But this is pretty neat.
- This by itself is kind of a fun little toy and hopefully you
- have enough information to do it yourself, or maybe
- experiment with more.
- What happens if you have more than n?
- Actually, maybe I'll do that in the next video.
- Anyway, see you soon.
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