Chapter 17 The binomial model of probability Part

Chapter 17: The binomial model of probability Part 3 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 pkqn-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 3 x – 2 x +y 4. 3 – a x+π 3. 4 e +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: 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)=(by the distributive property) x(x+y)+y(x+y) = x 2+ (xy+xy) +y 2 = x 2+2 xy+y 2 • The important thing to notice is that we actually have FOUR (4) terms when we expand a binomial 12

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), 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+x 2 y 2++y 1 x 2+y 1 y 2 • Let’s now set x=x 1=x 2, y=y 1=y 2 and substitute: xx+xy+xy+yy=x 2+2 xy+y 2 13

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? (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: The trinomial case • 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 and y 3 to track individual terms • So (x+y)3 becomes (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 =x 3 and y=y 1= y 2 =y 3 15

The binomial model: Expanding the trinomial • We have (x 1+y 1)(x 2 +y 2)(x 3+y 3) • Expanding out the first two terms, we get (x 1 x 2+x 2 y 2++y 1 x 2+y 1 y 2)(x 3+y 3)= 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 • 8 (23) 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 + xyy + yyy (3) x 3 + 3 x 2 y + 3 xy 2 + y 3 (4) 16

The binomial model: Firsts, squares and cubes • 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 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: 2 xy • After combining the terms, we get x 2+2 xy+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 + 3 x 2 y + 3 xy 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 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: 1 x 6 + 6 x 5 y + 15 x 4 y 2 +20 x 3 y 3 +15 x 2 y 4 +6 xy 5 + 1 y 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: Calculate the probabilities • 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! 40