Chapter 17: The binomial model of probability Part 3

1. Chapter 17:The binomial model of probabilityPart 3 AP Statistics

2. 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.

3. 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)

4. 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.

5. 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

6. 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.

7. 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.)

8. 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.

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., x1+4 or x1-y1)

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) = 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

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 (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

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? (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!)

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 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

15. 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)

16. The binomial model:Firsts, squares and cubes • So let’s review and see if there’s any kind of pattern we can find.

17. The binomial model: • If we take out the coefficients from each term, we get a table that looks like this (Pascal’s triangle):

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: • 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.

23. 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

24. 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

25. The binomial model:Expansion of the quadratic using tree diagram

26. 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

27. 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.

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

29. The binomial model:Cubics: multiply out every x and y

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

31. The binomial model:Cubics: grouping like terms

32. 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

33. 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:

34. 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:

35. 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!

36. 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:

37. The binomial model:Calculate the probabilities • So we get the following:

38. 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!