AP Statistics

Binomial model: tying it all together
Review of what we’ve already done
• Today, I want to show you how the binomial formulas we’ve been working with are related to, well, binomials as well as to the tree diagrams we’ve been doing.
• Hopefully it will all tie together for you and make sense.
• But first, some review. Somebody go to the board and write the formulas for the mean and standard deviation for a geometric model.
• When you’ve posted it and agree, go on to the next slide to see if you’ve gotten in right.
2

Binomial model: tying it all together
Review of what we’ve already done (2)
• Your answers should be:
Mean:
Standard deviation:
• Now, what are the standard deviation and the mean for the binomial model of probability? (see next slide for answer, after writing it on the board)
3

Binomial model: tying it all together
Review of what we’ve already done (3)
• Your answers should be:
Mean:
Standard deviation:
• Now, what is the formula for calculating the probabilities of the binomial distribution using the binomial coefficient? Express in terms of n, k, p and q. Write it on the board and go to the next slide.
4

Binomial model: tying it all together
Review of what we’ve already done (4)
• This is the formula we were working with yesterday. Be sure to remember it!
• Final question: write the formula for the binomial coefficient (aka the number of combinations possible for p k q n-k ). Write it on the whiteboard and check ur answer on next slide
5

Binomial model: tying it all together
Review of what we’ve already done (5)
• That’s right (at least I sure hope you got it right!):
• OK, ‘nuff review. Let’s start by showing you how what we’re doing relates to the expansion of binomials.
6

Binomial model/expanding binomials
What is a binomial?
• Review from pre-algebra/Algebra 1: what’s a binomial?
• Answer: a polynomial with two terms.
• TERRIBLE answer! My response:
7

Binomial model/expanding binomials
What is a binomial?(1)
• Review from pre-algebra/Algebra 1: what’s a binomial?
• Answer: a polynomial with two terms.
• TERRIBLE answer! My response:
• (Go to the next slide for a better answer.)
8

Binomial model: tying it all together
What is a binomial?(3)
• Either one variable and a constant or two variables, separated by an addition or subtraction sign so that there are, in fact, two terms
• Each term of the binomial can have a numeric multiple, including fractions (i.e., division) and
(which typically we don’t write)
• Spend 3 minutes and come up with 5 examples of binomials. Share out between tables, and discuss any disagreements. Examples on the next slide.
9

Binomial model: tying it all together
What is a binomial? (examples)
• Here are my examples
• How do they compare to yours?
• As always, YMMV.
• x+1
• 3x – 2
• x + y
• 4.3 – a
•
• x + π
• 3.4e +y
10

Binomial model: tying it all together
What is a binomial? (summary)
• 2 terms
• Separated by + or – (addition or subtraction)
• Can have coefficients
• Can have 1 or 2 variables
• Variables can only have the exponent of 1
(e.g., x 1 +4 or x 1 -y 1 )
11

The binomial model:
2
• Let’s approach the binomial problem by looking at what happens when we multiply out a binomial
• Lets start with expanding (x+y) 2
• (x+y) 2 = (x+y)(x+y)=(by the distributive property) x(x+y)+y(x+y) = x 2 + (xy+xy) +y 2 = x 2 +2xy+y 2
• The important thing to notice is that we actually have FOUR (4) terms when we expand a binomial
12

The binomial model:
• It’s easier to see what we’re doing if we label each factor as unique
• So, instead of (x+y)(x+y), let’s write the multiplication problem as (x
1
+y
1
)(x
2
+y
2
)
• Expanding as before, we get: x
1
(x
2
+y
2
) +y
1
(x
2
+y
2
)=x
1 x
2
• Let’s now set x=x
1
=x
2
, y=y
1
=y
2
+x
2 y
2+
+y
1 x
2
+y
1 y
2 and substitute: xx+xy+xy+yy=x 2 +2xy+y 2
13

The binomial model:
• Good question, and an important question. Hang in there for a bit.
• How many terms did we get when we expanded the binomial?
– 4, of which 2 (the xy-terms) were alike, so we combined them.
– How do the number of unique terms relate to the exponent? (2 n , where n=exponent)
• Now let’s do a cube to see if we can discover a pattern. (Math is more about patterns than numbers, in case you haven’t noticed!)
14

The binomial model:
• Same as with (x+y) 2 , except now it’s (x+y) 3
• We’re also going to use x
1
, y
1
, x
2
, y
2
, x
3 to track individual terms and y
3
• So (x+y) 3 becomes (x+y)(x+y)(x+y), which we’ll write as (x
1
+y
1
)(x
2
+y
2
)(x
3
+y
3
)
• We can do this simply by setting x= x
1
=x
2 and y=y
1
= y
2
=y
3
=x
3
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The binomial model:
• We have (x
1
+y
1
)(x
2
+y
2
)(x
3
+y
3
)
• Expanding out the first two terms, we get x
3 x
1 x
2
+x
3
(x x
2
1 y x
2
2
+x
+x
3 y
2
1 y x
2+
+y
2
+x
3 y
1
1 x y
2
2
+y
+y
1
3 x y
1
2 x
)(x
2
+y
3
3
+y x
2
3 y
2
)=
+y
3 y
1 x
2
+y
3 y
1 y
2
• 8 (2 3 ) terms; here’s how you simplify by substituting x and y back in to each term: x
3 x
1 x
2
+x
3 x
2 y
2
+x
3 y
1 x
2
+x
3 y
1 y
2
+y
3 x
1 x
2
+y
3 x
2 y
2
+y
3 y
1 x
2
+y
3 y
1 y
2
(1) xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy (circles=like terms) (2) xxx + xxy + xxy + xxy + xyy + xyy + xyy + yyy (3) x 3 + 3x 2 y + 3xy 2 + y 3 (4)
16

The binomial model:
• So let’s review and see if there’s any kind of pattern we can find.
• Here are the expansions for n=1 through n=5:
17

The binomial model:
• If we take out the coefficients from each term, we get a table that looks like this:
18

The binomial model:
• You can generate the triangle by expanding the 1’s down the outside and adding together the 2 numbers immediately above the entry:
19

The binomial model:
The first twelve rows of Pascal’s triangle
20

The binomial model:
Binomial coefficients are the entries
• Don’t believe that the binomial coefficients are involved? Look at the table this way:
21

The binomial model:
So what’s the big deal?
• Talk among yourselves and determine what the rule is for generating the blue numbers:
22

The binomial model:
23

The binomial model:
• Answer SHOULD be 2 n
• But what does that mean?
• It means that if you have (x+y) n , you will have n different permutations when you expand the binomial n times
• But we only want the number of COMBINATIONS, because in algebra xxy, xyx, and yxx are all the same things.
• Let’s show how this works in a 2-level tree diagram.
24

The binomial model:
Remembering the tree model
• The diagram at the right was one we did on refurbished computers
• Each branch has the probabilities
• We calculate the end probabilities by multiplying out all the branches together.
• We do the same thing with the binomial equation
25

The binomial model:
2-level tree diagram (the tree)
• Remember that each diagram has two branches coming off of each branch
• So a 2-level diagram should look like the diagram on the right
• We’re going to add x and y to each of the branches
26

The binomial model:
Expansion of the quadratic using tree diagram
27

The binomial model:
Summarizing the quadratic (n=2)
• 4 terms: x 2 , xy, yx, y 2
• xy and yx are the same term, so we combine them: 2xy
• After combining the terms, we get x 2 +2xy+y 2
• Adding the coefficients— 1 2 1 — and you get the total number of permutations
28

The binomial model:
Tree diagrams applied to cubes
• Just to get the pattern of what’s going on, let’s take a look at cubic equations and tree diagrams
• That is, the expansion of (x+y) 3 , which you will recall (I hope!) results in x 3 + 3x 2 y + 3xy 2 + y 3
• I will do this step by step.
29

The binomial model:
Cubics: put on the “probabilities” x and y
30

The binomial model:
Cubics: multiply out every x and y
31

The binomial model:
Cubics: multiply out the cubes of x and y
32

The binomial model:
Cubics: grouping like terms
33

The binomial model:
Things to remember
• For degree n polynomials, you will generate 2 n terms, i.e., permutations (i.e., for an 6 th -degree polynomial [x 6 ], you will general 2 6 (64) different terms)
• However, you will only have n+1 different terms (i.e., combinations)
– Using the (x+y) 6 , for example, you have 7 terms:
1x 6 + 6x 5 y + 15x 4 y 2 +20x 3 y 3 +15x 2 y 4 +6xy 5 + 1y 6
34

The binomial model:
Linking the binomial coefficient to the expansion
• Using a 6 th -order polynomial as an example, here’s how you connect the binomial coefficients with the equation:
35

The binomial model:
How to apply (using 6 th degree polynomial)
• You want to find the probability of 4 successes and one failure. Ignore for now the distribution between p and q
• n=6, k=5, so apply the equation:
36

The binomial model:
Example of how to apply binomial model
• Let’s take the model of the Olympic archer, who hit the bull’s-eye 80% of the time (this is
not a person you want to irritate!)
• p=0.8; q=0.2
• What is the probability that she will get 12 bull’s-eye in 15 shots?
• You do NOT want to be calculating the permutations on this one by hand!
37

The binomial model:
12 bull’s-eyes out of 15 shots
• We get the number of combinations of 12 out of 5 by calculating the binomial coefficient:
38

The binomial model:
• So we get the following:
39

The binomial model:
The formula works better than Pascal’s triangle
• Oh, yes, it does! Here’s what you’d have to do for the triangle…and this is only the 16 th row!
(n k) 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 q n-k
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 n p 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
40