Probability Distributions and Expected Value

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Probability Distributions and Expected Value

• Chapter 5.1 Probability Distributions and
  Predictions
• Mathematics of Data Management (Nelson)
• MDM 4U
• Authors Gary Greer (with K. Myers)

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Probability Distributions of a Discrete Random
Variable

• a discrete random variable is one that can take
  on only a finite number of values
• for example, rolling a die can only produce
  numbers in the set 1,2,3,4,5,6
• rolling 2 dice can produce only numbers in the
  set 2,3,4,5,6,7,8,9,10,11,12
• choosing a card from a complete deck can produce
  only the cards in the set A,2,3,4,5,6,7,8,9,10,J,
  Q,K

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Probability Distribution

• a probability distribution of a random variable
  x, is a function which provides the probability
  of each possible value of x
• this function may be represented as a table of
  values, a graph or a mathematical expression
• for example, rolling a die

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Probability Distribution for 2 Dice
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What would a probability distribution graph for
three dice look like?

• lets try it! Using three dice, figure out how
  many possible cases there are
• now find out how many possible ways there are to
  create each of the possible cases
• fill in a table like the one below
• now you can make your graph

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So what does an experimental distribution look
like?

• a simulated dice throw was done a million times
  using a Java program and generated the following
  data
• what is the most common outcome?
• does this make sense?

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Back to 2 Dice

• What is the expected value of throwing 2 dice?
• How could this be calculated?
• So the expected value of a discrete variable x is
  the sum of the values of x multiplied by their
  probabilities

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Example tossing 3 coins

• what is the likelihood of at least 2 heads?
• it must be the total probability of tossing 2
  heads and tossing 3 heads
• P(X 2) P(X 3) ? ? ½
• so the probability is 0.5

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Example tossing 3 coins

• what is the expected number of heads
• it must be the sums of the values of x multiplied
  by the probabilities of x
• 0P(X 0) 1P(X 1) 2P(X 2) 3P(X 3)
• 0(?) 1(?) 2(?) 3(?) 1½
• so the expected number of heads is 1.5

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Selecting a Team of three people from a group of
4 men and 3 women

• what is the probability of having at least one
  woman on the team?
• there are C(7,3) or 35 possible teams
• C(4,3) have no women
• C(4,2) x C(3,1) have one woman
• C(4,1) x C(3,2) have 2 women
• C(3,3) have 3 women

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Example selecting a committee

• what is the likelihood of at least one woman?
• it must be the total probability of all the cases
  with at least one woman
• P(X 1) P(X 2) P(X 3)
• 18/35 12/35 1/35 31/35
• is there an easier way????

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Example selecting a committee

• what is the expected number of women?
• 0P(X 0) 1P(X 1) 2P(X 2) 3P(X 3)
• 0(4/35) 1(18/35) 2(12/35) 3(1/35)
• 1.3 (approximately)

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Exercises / Homework

• Homework
• page 277 1, 2, 3, 4, 5, 9, 12, 13

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Pascals Triangle and the Binomial Theorem

• Chapter 5.2 Probability Distributions and
  Predictions
• Mathematics of Data Management (Nelson)
• MDM 4U
• Authors Gary Greer (with K. Myers)

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How many routes are there to the top right-hand
corner?

• you need to move up 4 spaces and over 5 spaces
• the total routes can be calculated with C(9,5) or
  C(9,4)
• 126 ways

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The Binomial Theorem

• the term (a b) can be expanded
• (a b)0 1
• (a b)1 a b
• (a b)2 a2 2ab b2
• (a b)3 a3 3a2b 3ab2 b3
• (a b)4 a4 4a3b 6a2b2 4ab3 b4
• Blaise Pascal (for whom the Pascal computer
  language is named) noted that there are patterns
  of expansion, and from this he developed what we
  now know as Pascals Triangle. He also invented
  the second mechanical calculator

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Pascals Triangle

• the outer values are always 1
• the inner values are determined by adding the
  values of the two values diagonally above

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1

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Pascals Triangle

• sum of each row is
• 1 20
• 2 21
• 4 22
• 8 23
• 16 24
• 32 25
• 64 26

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 1 6 15 20 15 6 1

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Pascals Triangle

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 1 6 15 20 15 6 1

• Uses?
• binomial theorem
• combinations!
• choose 2 items from 5
• go to the 5th row, the 2nd number 10 (always
  start counting at 0)
• so it can be used to find combinations
• modeling the electrons in each shell of an atom

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Pascals Triangle Cool Stuff

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 1 6 15 20 15 6 1

• each diagonal is summed up in the next value
  below and to the left
• called the hockey stick property

• there may even be music hidden in it
• http//www.geocities.com/Vienna/9349/pascal.mid

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Pascals Triangle Cool Stuff

• numbers divisible by 5
• similar patterns exist for other numbers
• http//www.shodor.org/interactivate/activities/pas
  cal1/

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Pascals Triangle can also be seen in terms of
combinations

• n 0
• n 1
• n 2
• n 3
• n 4
• n 5
• n 6

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Pascals Triangle

• symmetrical down the middle
• outside is always 1
• second diagonal values match the row numbers
• sum of each row is a power of 2
• sum of nth row is 2n
• number inside a row is the sum of the two numbers
  above it

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So what does this have to do with the Binomial
Theorem

• remember that
• (a b)4 a4 4a3b 6a2b2 4ab3 b4
• and the triangles 4th row is 1 4 6 4 1
• so Pascals Triangle allows you to predict the
  coefficients in the binomial expansion
• notice also that the exponents on the variables
  also form a predictable pattern with the
  exponents of each term having a sum of n

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The Binomial Theorem
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A Binomial Expansion

• lets expand (x y)4

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Another Binomial Expansion

• lets expand (a 4)5

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Some Binomial Examples

• what is the 6th term (a b)9?
• dont forget that when you find the 6th term, r
  5
• what is the 11th term of (2x 4)12

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Finding the term
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Finding the term
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Look at the triangle in a different way

• r0 r1 r2 r3 r4 r5
• n 0 1
• n 1 1 1
• n 2 1 2 1
• n 3 1 3 3 1
• n 4 1 4 6 4 1
• n 5 1 5 10 10 5 1
• n 6 1 6 15 20 15 6 1

• for a binomial expansion of
• (a b)5, the term for r 3 has a coefficient of
  10

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And one more thing

• remember that for the inner numbers in the
  triangle, any number is the sum of the two
  numbers above it
• for example 4 6 10
• this suggests the following
• which provides an example of Pascals Identity

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For Example
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How can this help us solve our original problem?

• so by overlaying Pascals Triangle over the grid
  we can see that there are 126 ways to move from
  one corner to another

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How many routes pass through the green square?

• to get to the green square, there are C(4,2) ways
  (6 ways)
• to get to the end from the green square there are
  C(5,3) ways (10 ways)
• in total there are 60 ways

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How many routes do not pass through the green
square?

• there are 60 ways that pass through the green
  square
• there are C(9,5) or 126 ways in total
• then there must be 126 60 paths that do not
  pass through the green square

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Exercises / Homework

• Homework read the examples on pages 281-287, in
  particular the example starting on the bottom of
  page 287 is important
• page 289
• 1, 2 a c e g, 3, 4, 5, 6, 8, 9, 11, 13