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### Chapter 17:The binomial model of probabilityPart 3

AP Statistics

Binomial model: tying it all togetherReview 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.

Binomial model: tying it all togetherReview 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)

Binomial model: tying it all togetherReview 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.

Binomial model: tying it all togetherReview 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 pkqn-k). Write it on the whiteboard and check ur answer on next slide

Binomial model: tying it all togetherReview 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.

Binomial model/expanding binomialsWhat 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.)

Binomial model: tying it all together What is a binomial?(2)

- 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.

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

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., x1+4 or x1-y1)

The binomial model:Example using (x+y)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) = x2+ (xy+xy) +y2 = x2+2xy+y2
- The important thing to notice is that we actually have FOUR (4) terms when we expand a binomial

The binomial model:Tracking the members of a binomial

- 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 (x1+y1)(x2+y2)
- Expanding as before, we get:

x1 (x2+y2) +y1 (x2+y2)=x1 x2+x1y2++y1x2+y1y2

- Let’s now set x=x1=x2, y=y1=y2 and substitute:

xx+xy+xy+yy=x2+2xy+y2

The binomial model:So what?

- 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? (2n, 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!)

The binomial model:The trinomial case

- Same as with (x+y)2, except now it’s (x+y)3
- We’re also going to use x1, y1, x2, y2, x3 and y3 to track individual terms
- So (x+y)3 becomes (x+y)(x+y)(x+y), which we’ll write as (x1+y1)(x2 +y2)(x3+y3)
- We can do this simply by setting x= x1=x2 =x3 and y=y1= y2 =y3

The binomial model:Expanding the trinomial

- We have (x1+y1)(x2 +y2)(x3+y3)
- Expanding out the first two terms, we get

(x1x2+x1y2++y1x2+y1y2)(x3+y3)=

x3x1x2+x3x2y2+x3y1x2+x3y1y2+y3x1x2+y3x2y2+y3y1x2+y3y1y2

- 8 (23) terms; here’s how you simplify by substituting x and y back in to each term:

x3x1x2+x3x2y2+x3y1x2+x3y1y2+y3x1x2+y3x2y2+y3y1x2+y3y1y2 (1)

xxx + xxy + xyx + xyy + yxx + yxy + yyx + yyy (circles=like terms) (2)

xxx + xxy + xxy + xxy + xyy + xyy + xyy + yyy (3)

x3 + 3x2y + 3xy2 + y3 (4)

The binomial model:Firsts, squares and cubes

- So let’s review and see if there’s any kind of pattern we can find.

The binomial model:

- If we take out the coefficients from each term, we get a table that looks like this (Pascal’s triangle):

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:

The binomial model:Binomial coefficients are the entries

- Don’t believe that the binomial coefficients are involved? Look at the table this way:

The binomial model:So what’s the big deal?

- Talk among yourselves and determine what the rule is for generating the blue numbers:

The binomial model:

- Answer SHOULD be 2n
- 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.

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

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

The binomial model:Summarizing the quadratic (n=2)

- 4 terms: x2, xy, yx, y2
- xy and yx are the same term, so we combine them: 2xy
- After combining the terms, we get x2+2xy+y2
- Adding the coefficients— 1 2 1 — and you get the total number of permutations

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 x3 + 3x2y + 3xy2 + y3
- I will do this step by step.

The binomial model:Things to remember

- For degree n polynomials, you will generate 2n terms, i.e., permutations (i.e., for an 6th-degree polynomial [x6], you will general 26 (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:

1x6 + 6x5y + 15x4y2 +20x3y3 +15x2y4 +6xy5 + 1y6

The binomial model:Linking the binomial coefficient to the expansion

- Using a 6th-order polynomial as an example, here’s how you connect the binomial coefficients with the equation:

The binomial model:How to apply (using 6th degree polynomial)

- You want to find the probability of 4 successes and 2 failures. Ignore for now the distribution between p and q
- n=6, k=4, so apply the equation:

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-eyes in 15 shots?
- You do NOT want to be calculating the permutations on this one by hand!

The binomial model:12 bull’s-eyes out of 15 shots

- We get the number of combinations of 12 out of 15 by calculating the binomial coefficient:

The binomial model:Calculate the probabilities

- So we get the following:

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 16th row!

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