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Chapter 17: The binomial model of probability Part 3. AP Statistics. Binomial model: tying it all together Review of what we’ve already done.

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binomial model tying it all together review of what we ve already done
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.
binomial model tying it all together review of what we ve already done 2
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)
binomial model tying it all together review of what we ve already done 3
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.
binomial model tying it all together review of what we ve already done 4
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
binomial model tying it all together review of what we ve already done 5
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.
binomial model expanding binomials what is a binomial 1
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.)

binomial model tying it all together what is a binomial 2
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.
binomial model tying it all together what is a binomial examples
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
binomial model tying it all together what is a binomial summary
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)
the binomial model example using x y 2
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
the binomial model tracking the members of a binomial
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

the binomial model so what
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!)
the binomial model the trinomial case
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
the binomial model expanding the trinomial
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)

the binomial model firsts squares and cubes
The binomial model:Firsts, squares and cubes
  • So let’s review and see if there’s any kind of pattern we can find.
the binomial model
The binomial model:
  • If we take out the coefficients from each term, we get a table that looks like this (Pascal’s triangle):
the binomial model1
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:
the binomial model binomial coefficients are the entries
The binomial model:Binomial coefficients are the entries
  • Don’t believe that the binomial coefficients are involved? Look at the table this way:
the binomial model so what s the big deal
The binomial model:So what’s the big deal?
  • Talk among yourselves and determine what the rule is for generating the blue numbers:
the binomial model2
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.
the binomial model remembering the tree model
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
the binomial model 2 level tree diagram the tree
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
the binomial model summarizing the quadratic n 2
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
the binomial model tree diagrams applied to cubes
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.
the binomial model things to remember
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

the binomial model linking the binomial coefficient to the expansion
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:
the binomial model how to apply using 6 th degree polynomial
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:
the binomial model example of how to apply binomial model
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!
the binomial model 12 bull s eyes out of 15 shots
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:
the binomial model the formula works better than pascal s triangle
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!