Chapter 17:The binomial model of probability

Part 3

AP Statistics

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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.

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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)

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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.

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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

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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.

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Binomial model/expanding binomialsWhat is a binomial?

• Review from pre-algebra/Algebra 1: what’s a binomial?

• Answer: a polynomial with two terms.• TERRIBLE answer! My response:

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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.)

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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.

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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•

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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)

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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

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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+x2y2++y1x2+y1y2

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

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

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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!)

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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

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The binomial model:Expanding the trinomial

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

(x1x2+x2y2++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)

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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:

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The binomial model:

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

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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:

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

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The binomial model:Binomial coefficients are the entries

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

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

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

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The binomial model:

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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.

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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

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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

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The binomial model:Expansion of the quadratic using tree diagram

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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

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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.

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

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

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

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The binomial model:Cubics: grouping like terms

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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

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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:

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The binomial model:How to apply (using 6th 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:

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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!

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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:

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The binomial model:Calculate the probabilities

• So we get the following:

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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!

(n k) 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1 qn-k 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 pn 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15