Chapter 17: The binomial model of probability Part 3

Chapter 17:The binomial model of probabilityPart 3AP StatisticsBinomial model: tying it all togetherReview of what weve already doneToday, I want to show you how the binomial formulas weve been working with are related to, well, binomials as well as to the tree diagrams weve 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 youve posted it and agree, go on to the next slide to see if youve gotten in right.2

Binomial model: tying it all togetherReview of what weve 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 weve 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 weve 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 weve already done (5)Thats right (at least I sure hope you got it right!):

OK, nuff review. Lets start by showing you how what were doing relates to the expansion of binomials.

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Binomial model/expanding binomialsWhat is a binomial?(1)Review from pre-algebra/Algebra 1: whats a binomial?Answer: a polynomial with two terms.TERRIBLE answer! My response:(Go to the next slide for a betteranswer.)

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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 termsEach term of the binomial can have a numeric multiple, including fractions (i.e., division) and (which typically we dont 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 examplesHow do they compare to yours?As always, YMMV.x+13x 2x + y4.3 ax + 3.4e +y

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Binomial model: tying it all together What is a binomial? (summary)2 termsSeparated by + or (addition or subtraction)Can have coefficientsCan have 1 or 2 variablesVariables can only have the exponent of 1 (e.g., x1+4 or x1-y1)10

The binomial model:Example using (x+y)2Lets approach the binomial problem by looking at what happens when we multiply out a binomialLets 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+y2The important thing to notice is that we actually have FOUR (4) terms when we expand a binomial11

11The binomial model:Tracking the members of a binomialIts easier to see what were doing if we label each factor as uniqueSo, instead of (x+y)(x+y), lets write the multiplication problem as (x1+y1)(x2+y2)Expanding as before, we get:x1 (x2+y2) +y1 (x2+y2)=x1 x2+x1y2++y1x2+y1y2Lets now set x=x1=x2, y=y1=y2 and substitute: xx+xy+xy+yy=x2+2xy+y212

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 lets do a cube to see if we can discover a pattern. (Math is more about patterns than numbers, in case you havent noticed!)13

The binomial model:The trinomial caseSame as with (x+y)2, except now its (x+y)3Were also going to use x1, y1, x2, y2, x3 and y3 to track individual termsSo (x+y)3 becomes (x+y)(x+y)(x+y), which well write as (x1+y1)(x2 +y2)(x3+y3)We can do this simply by setting x= x1=x2 =x3 and y=y1= y2 =y3 14

The binomial model:Expanding the trinomialWe 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; heres 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 cubesSo lets review and see if theres any kind of pattern we can find.

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The binomial model:

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

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The binomial model:

You can generate the triangle by expanding the 1s down the outside and adding together the 2 numbers immediately above the entry:18

The binomial model:The first twelve rows of Pascals triangle

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The binomial model:Binomial coefficients are the entriesDont believe that the binomial coefficients are involved? Look at the table this way:20

The binomial model:So whats the big deal?Talk among yourselves and determine what the rule is for generating the blue numbers:21

The binomial model:

Answer SHOULD be 2nBut 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 timesBut we only want the number of COMBINATIONS, because in algebra xxy, xyx, and yxx are all the same things.Lets show how this works in a 2-level tree diagram.22

The binomial model:Remembering the tree modelThe diagram at the right was one we did on refurbished computersEach branch has the probabilitiesWe 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 branchSo a 2-level diagram should look like the diagram on the rightWere going to add x and y to each of the branches24

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, y2xy and yx are the same term, so we combine them: 2xyAfter combining the terms, we get x2+2xy+y2Adding the coefficients 1 2 1 and you get the total number of permutations

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The binomial model:Tree diagrams applied to cubesJust to get the pattern of whats going on, lets take a look at cubic equations and tree diagramsThat is, the expansion of (x+y)3, which you will recall (I hope!) results in x3 + 3x2y + 3xy2 + y3I will do this step by step.27

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 rememberFor 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 + 1y632

The binomial model:Linking the binomial coefficient to the expansion Using a 6th-order polynomial as an example, heres 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 2 failures. Ignore for now the distribution between p and qn=6, k=4, so apply the equation:34

The binomial model:Example of how to apply binomial modelLets take the model of the Olympic archer, who hit the bulls-eye 80% of the time (this is not a person you want to irritate!)p=0.8; q=0.2What is the probability that she will get 12 bulls-eyes in 15 shots?You do NOT want to be calculating the permutations on this one by hand!35

The binomial model:12 bulls-eyes out of 15 shotsWe get the number of combinations of 12 out of 15 by calculating the binomial coefficient:

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The binomial model:Calculate the probabilitiesSo we get the following:37

The binomial model:The formula works better than Pascals triangleOh, yes, it does! Heres what youd have to do for the triangleand this is only the 16th row! 38

(n k)11510545513653003500564356435500530031365455105151

qn-k1514131211109876543210

pn0123456789101112131415