4.1 (cont.) Probability Models

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4.1 (cont.) Probability Models

• The Equally Likely Approach
• (also called the Classical Approach)

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Assigning Probabilities

• If an experiment has N outcomes, then each
  outcome has probability 1/N of occurring
• If an event A1 has n1 outcomes, then
• P(A1) n1/N

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Dice You toss two dice. What is the probability
of the outcomes summing to 5?
This is S (1,1), (1,2), (1,3), etc.
There are 36 possible outcomes in S, all equally
likely (given fair dice). Thus, the probability
of any one of them is 1/36. P(the roll of two
dice sums to 5) P(1,4) P(2,3) P(3,2)
P(4,1) 4 / 36 0.111
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We Need Efficient Methods for Counting Outcomes
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Product Rule for Ordered Pairs

• A student wishes to commute to a junior college
  for 2 years and then commute to a state college
  for 2 years. Within commuting distance there are
  4 junior colleges and 3 state colleges. How many
  junior college-state college pairs are available
  to her?

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Product Rule for Ordered Pairs

• junior colleges 1, 2, 3, 4
• state colleges a, b, c
• possible pairs
• (1, a) (1, b) (1, c)
• (2, a) (2, b) (2, c)
• (3, a) (3, b) (3, c)
• (4, a) (4, b) (4, c)

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Product Rule for Ordered Pairs

• junior colleges 1, 2, 3, 4
• state colleges a, b, c
• possible pairs
• (1, a) (1, b) (1, c)
• (2, a) (2, b) (2, c)
• (3, a) (3, b) (3, c)
• (4, a) (4, b) (4, c)

4 junior colleges 3 state colleges total number
of possible pairs 4 x 3 12
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Product Rule for Ordered Pairs

• junior colleges 1, 2, 3, 4
• state colleges a, b, c
• possible pairs
• (1, a) (1, b) (1, c)
• (2, a) (2, b) (2, c)
• (3, a) (3, b) (3, c)
• (4, a) (4, b) (4, c)

In general, if there are n1 ways to choose the
first element of the pair, and n2 ways to
choose the second element, then the number of
possible pairs is n1n2. Here n1 4, n2 3.
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Counting in Either-Or Situations

• NCAA Basketball Tournament, 68 teams how many
  ways can the bracket be filled out?
• How many games?
• 2 choices for each game
• Number of ways to fill out the bracket
• 267 1.5 1020
• Earth pop. about 6 billion everyone fills out
  100 million different brackets
• Chances of getting all games correct is about 1
  in 1,000

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A states automobile license plate begins with a
number from 1 to 26, corresponding to the 26
counties in a state. This number is followed by
a 5-digit number. How many different license
plates can the state issue?

• 1,300
• 6,552
• 2,600,000
• 786,240
• 26,000

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Counting Example

• Pollsters minimize lead-in effect by rearranging
  the order of the questions on a survey
• If Gallup has a 5-question survey, how many
  different versions of the survey are required if
  all possible arrangements of the questions are
  included?

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Solution

• There are 5 possible choices for the first
  question, 4 remaining questions for the second
  question, 3 choices for the third question, 2
  choices for the fourth question, and 1 choice for
  the fifth question.
• The number of possible arrangements is therefore
• 5 ? 4 ? 3 ? 2 ? 1 120

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Efficient Methods for Counting Outcomes

• Factorial Notation
• n!1?2? ?n
• Examples
• 1!1 2!1?22 3! 1?2?36 4!24
• 5!120
• Special definition 0!1

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Factorials with calculators and Excel

• Calculator
• non-graphing x ! (second function)
• graphing bottom p. 9 T I Calculator Commands
• (math button)
• Excel
• Insert function Math and Trig category, FACT
  function

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Factorial Examples

• 20! 2.43 x 1018
• 1,000,000 seconds?
• About 11.5 days
• 1,000,000,000 seconds?
• About 31 years
• 31 years 109 seconds
• 1018 109 x 109
• 20! is roughly the age of the universe in seconds

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Permutations

• A B C D E
• How many ways can we choose 2 letters from the
  above 5, without replacement, when the order in
  which we choose the letters is important?
• 5 ? 4 20

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Permutations (cont.)
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Permutations with calculator and Excel

• Calculator
• non-graphing nPr
• Graphing
• p. 9 of T I Calculator Commands
• (math button)
• Excel
• Insert function Statistical, Permut

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Combinations

• A B C D E
• How many ways can we choose 2 letters from the
  above 5, without replacement, when the order in
  which we choose the letters is not important?
• 5 ? 4 20 when order important
• Divide by 2 (5 ? 4)/2 10 ways

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Combinations (cont.)
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BUS/ST 350 Powerball Lottery
From the numbers 1 through 20, choose 6 different
numbers. Write them on a piece of paper.
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Chances of Winning?
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Example Illinois State Lottery
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North Carolina Powerball Lottery

• Prior to Jan. 1, 2009

• After Jan. 1, 2009

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The Forrest Gump Visualization of Your Lottery
Chances

• How large is 195,249,054?
• 1 bill and 100 bill both 6 in length
• 10,560 bills 1 mile
• Lets start with 195,249,053 1 bills and one
  100 bill
• and take a long walk, putting down bills
  end-to-end as we go

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Raleigh to Ft. Lauderdale
still plenty of bills remaining, so continue
from
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Ft. Lauderdale to San Diego
still plenty of bills remaining, so continue
from
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San Diego to Seattle
still plenty of bills remaining, so continue
from
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Seattle to New York
still plenty of bills remaining, so continue
from
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New York back to Raleigh
still plenty of bills remaining, so
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Go around again! Lay a second path of bills
Still have 5,000 bills left!!
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Chances of Winning NC Powerball Lottery?

• Remember one of the bills you put down is a 100
  bill all others are 1 bills.
• Put on a blindfold and begin walking along the
  trail of bills.
• Your chance of winning the lottery the chance of
  selecting the 100 bill if you stop at a random
  location along the trail and pick up a bill .

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Virginia State Lottery