Counting Techniques (Dr. Monticino)

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Counting Techniques (Dr. Monticino)
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Overview

• Why counting?
• Counting techniques
• Multiplication principle
• Permutation
• Combination
• Examples
• Probability examples

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Why Counting?

• Recall that if each outcome of an experiment is
  assumed to be equally likely, then the
  probability of an event is k/n
• where k is the number of elements in the event
  and n is the number of elements in the sample
  space
• So to calculate the probability of an event, we
  need to be able to count the number of elements
  in the event and in the sample space

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Multiplication Principle

• Multiplication principle. Suppose that an
  experiment can be regarded as a series of k
  sub-experiments. Such that the first
  sub-experiment has n1 possible outcomes, the
  second sub-experiment has n2 possible outcomes,
  and so on. Then the total number of outcomes in
  the main experiment is
• n1 x n2 x ... x nk
• Examples
• Flip a coin and roll a die
• Roll 5 die or roll a single die five times

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Permutation

• Factorial. n! (read n factorial) equals
• Permutation. The number of ways to select r
  objects, in order, out of n objects equals

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Examples

• How many ways are there to do the following
• Line up 10 people
• Select a President, VP and Treasurer from a group
  of 10 people
• Sit 5 men and 5 women in a row, alternating gender

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Combination

• Combination. The number of ways to select r
  objects out of n objects when order is not
  relevant equals

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Examples

• How many ways are there to do the following
• Select 3 people from a group of 10
• Select 7 people from a group of 10
• Get exactly 5 heads out of 12 coin flips

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Probability Examples

• Select three people at random from a group of 5
  women and and 5 men
• What is the probability that all those selected
  are men?
• What is the probability that at least one women
  is chosen?
• What is the probability that at least two women
  are chosen?

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Probability Examples

• Flip a fair coin 3 times
• What is the probability that 3 heads come up?
• What is the probability that at least 1 tail
  occurs?
• What is the probability that exactly 2 tails
  occur?
• What is the probability that at least 2 tails
  occur?

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Probability Examples

• Play roulette 3 times
• What is the probability that red comes up every
  time?
• What is the probability that black comes up at
  least once?
• What is the probability that black comes up
  exactly two times?
• What is the probability that black comes up at
  least two times?

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Probability Examples

• Flip a fair coin 10 times
• What is the probability that 10 heads come up?
• What is the probability that at least 1 tail
  occurs?
• What is the probability that exactly 8 tails
  occur?
• What is the probability that at least 8 tails
  occur?

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Probability Examples

• Play roulette 20 times
• What is the probability that red comes up every
  time?
• What is the probability that black comes up at
  least once?
• What is the probability that black comes up
  exactly 18 times?
• What is the probability that black comes up at
  least 18 times?

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Probability Examples

• Roll a fair die 5 times
• What is the probability that an ace comes up all
  five times?
• What is the probability that an ace occurs at
  least once?
• What is the probability that an ace occurs
  exactly 3 times?
• What is the probability that an ace occurs at
  least 3 times?

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Probability Examples

• To win the jackpot in Lotto Texas you need to
  match all six of the numbers drawn (5 numbers are
  selected from numbers 1 to 44 and the sixth is
  selected separately from 1 to 44)
• What is the probability of winning if you buy one
  ticket?
• What is the probability of winning if you buy
  five tickets?
• Is it better to buy five tickets in one Lotto
  drawing or a single ticket in five successive
  Lotto games?

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Assignment Sheet

• Read Chapter 15 carefully
• Redo all problems from lecture
• Not to turn in
• (Dr. Monticino)