Transcript Chapter 17: The binomial model of probability Part 3

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.
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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)
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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.
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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 pkqn-k). Write it on the
whiteboard and check ur answer on next slide
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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.
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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.)
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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 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+x1y2++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+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)
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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.
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The binomial model:
• If we take out the coefficients from each term,
we get a table that looks like this (Pascal’s
triangle):
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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:
• 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 2 failures. Ignore for now the distribution
between p and q
• n=6, k=4, 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-eyes 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 15 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
n-k
q 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
n
p 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
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