Introduction to Probability: Counting Methods

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Introduction to Probability Counting Methods

• Rutgers University
• Discrete Mathematics for ECE
• 14332202

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Why Probability?

• We can describe processes for which the outcome
  is uncertain
• By their average behavior
• By the likelihood of particular outcomes
• Allows us to build models for many physical
  behaviors
• Speech, images, traffic

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Applications

• Communications
• Speech and Image Processing
• Machine Learning
• Decision Making
• Network Systems
• Artificial Intelligence
• Used in many undergraduate courses (every grad
  course)

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Methods of Counting

• One way of interpreting probability is by the
  ratio of favorable to total outcomes
• Means we need to be able to count both the
  desired and the total outcomes
• For illustration, we explore only the most
  important applications
• Coin flipping
• Dice rolling
• Card Games

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Combinatorics

• Mathematical tools to help us count
• How many ways can 12 distinct objects be
  arranged?
• How many different sets of 4 objects be chosen
  from a group of 20 objects?
• -- Extend this to find probabilities

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Combinatorics

• Number of ways to arrange n distinct objects
• n!
• Number of ways to obtain an ordered sequence of k
  objects from a set of n n!/(n-k)! -- k
  permutation
• Number of ways to choose k objects out of n
  distinguishable objects

This one comes up a lot!
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Set Theory and Probability

• We use the same ideas from set theory in our
  study of probability
• Experiment
• Roll a dice
• Outcome any possible observation of an exp.
• Roll a six
• Sample Space the set of all possible outcomes
• 1,2,6
• Event set of outcomes
• Dice rolled is odd

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Venn Diagrams

• Outcomes are mutually exclusive disjoint

S
2
3
1
5
4
6
Event A
Outcomes
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An Example from Card Games

• What is the probability of drawing two of the
  same card in a row in a shuffled deck of cards?
• Experiment
• Pulling two cards from the deck
• Event Space
• All outcomes that describe our event
• Two cards are the same
• Sample Space
• All Possible Outcomes
• All combinations of 2 cards from a deck of 52

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Sample Space/Event Space

• Venn Diagram

Event Space (set of favorable outcomes)
S
all possible outcomes
A,A
K,2
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Calculating the Probability

• P(Event)
• Expressed as the ratio of favorable outcomes to
  total outcomes
• -- Only when all outcomes are EQUALLY LIKELY

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Probabilities from Combinations

• Rule of Product
• Total number of two card combinations?
• We need to find all the combinations of suit and
  value that describe our event set use rule of
  product to find the number of combinations
• First, we find number of values 13 choices, and
  choices of suits to give our number of
  possible outcomes ? 136 78
• Probability(Event) 78/1326 0.0588

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Probabilities from Subexperiments

• Only holds for independent experiments
• Lets look at the last problem
• Two subexperiments
• First can be anything 52/52 1
• Second, must be one of the 3 remaining cards of
  the same value from 51 remaining cards ? 3/51
  0.588

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An Example from Dice Rolling

• Experiment Roll Two (6-sided) Dice
• Event Numbers add to 7
• Sample Space (all possible outcomes)S

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Sample Space/Event Space
Event Space

• Venn Diagram

S
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Calculating Probability

• P(Event) 6/36 1/6

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Side Note

• Probability is something we calculate
  theoretically as a value between 0 and 1, it is
  not something calculated through experimentation
  (that is more statistics).
• Just because you roll a dice 100 times, and it
  came up as a 1 20 times, does not make P(roll a
  1) 0.2
• It would be the limiting case in doing an
  infinite number of experiments, but this is
  impossible.
• So, call your calculated values the
  probability, and your experimental values the
  relative frequency.