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 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: 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) = 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: 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 (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 +2xy+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)(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 (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: 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 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— — 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: 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