COSC 4350 Artificial Intelligence

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COSC 4350Artificial Intelligence

• Uncertain Reasoning Basics of Probability Theory
  (Part 2)
• Dr. Lappoon R. Tang

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Overview

• Joint probabilities
• Independent events
• Random variables
• Probability distributions
• Disjoint events

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Readings

• R N Chapter 13
• Sec 13.5

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Revision Basic Concepts

• The sample space is the set of all possible
  outcomes (in an experiment)
• Example in tossing two coins, the set of all
  possible outcomes (H,H), (H,T), (T,H), (T,T)
• An event is a subset of the sample space
• Example in tossing two coins, the event at
  least one head appears (H,H), (H,T), (T,H)
• The sample space itself is also an event!
• An event is a particular collection of outcomes

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Revision Basic Concepts (contd)

• Let S be the set of all possible outcomes (i.e.
  the sample space). The (unconditional or prior)
  probability P(A) (the chance that event A
  occurred, i.e. one of the outcomes of A happens)
  is given by
• Where A is the number of outcomes in A and S
  is the sample space

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Revision Basic Concepts (contd)

• Example the set of all possible outcomes in
  tossing two fair coins (H,H), (H,T), (T,H),
  (T,T), the event A two coins turn up the same
  side consists of the set of outcomes (H,H),
  (T,T)
• So, p(A) A / S 2 / 4 0.5

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Joint Probability probability of co-occurrence
of events

• The probability that event A co-occurs with event
  B is defined as
• P(A,B) of times A and B happen together / the
  total of combinations of events that can happen
• It measures the degree to which event A is
  associated with event B
• If P(A,B) 0, A and B never happen together
• If P(A,B) 1, A and B always happen together

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Joint Probability probability of co-occurrence
of events

• Example (in flipping two fair coins) the
  probability that the first coin shows a H and the
  second coin a T
• P(First coin H, Second coin T)
• To calculate this probability, you need to count
  the number of pairs (First Coin, Second Coin)
  such that the first H and second T and divide
  it by all possible number of pairs
• Answer 1/4
• General Case P(A1,A2,A3, ,An) the probability
  that the N events happen together

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Independent Events

• If two events A and B are independent of each
  other there is no casual link between the two
  (that A happens tells nothing about if B would
  happen), then P(A,B) P(A) x P(B)
• Example The probability that the first coin
  turns out a head and the second coin turns out a
  tail (assuming that the two coins are independent
  of each other) P(first coin is a H, second
  coin is a T) P(H) x P(T) ½ x ½ ¼

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Random Variables

• Writing out every outcome literally could be
  cumbersome, sometimes itd be nice to summarize
  an event by a number
• Example 160 coins were tossed, and you want to
  find the probability that 45 heads turned up,
  itd be too much trouble to have to express that
  event literally as a set of outcomes like
  HHH..TTTT, HTH..THHT,
• We can compactly represent an event by a variable
  X that denotes the number of heads in an outcome
• P(HHH TTTT, ) P(X 45)
• Technically, X is really a function
• X the power set of S (sample space) ? R (the
  real numbers)
• X P (S) ? R (a real number)
• X(event) r (a real number)

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Probability Distribution

• A probability distribution indicates the
  probabilities of various events (represented by a
  random variable) in the sample space
• Discrete probability distribution
• Simply a listing of probabilities of events like
  lt0.1, 0.5, 0.4gt (suppose there are only three
  events).
• Must sum to 1.
• Continuous probability distribution
• Something you dont need to know right now

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Fun Problem

• You are participating in a TV game show. You are
  shown three doors and are told that there is a
  car behind one door and a frog behind each of the
  other two doors.
• You are allowed to select one door (without
  opening it). The game show host opens one of the
  other doors, showing you a frog.
• Should you now pick the other closed door, or
  stick with your original pick?

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Fun Problem Answer

• It doesnt matter because the other frog is going
  to jump out through the opened door and be with
  his/her mate

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Exercise One Disjoint Events

• Two events A and B are disjoint if they dont
  share any outcome (A intersect B empty set)
• Suppose we are tossing three fair coins, let the
  event A HHH, HTH, THT, and the event B
  TTT, HHT, TTH.
• Compute the size of the sample space S
• Compute P(A)
• Compute P(B)
• Compute A U B
• Compute P(A U B)
• Compare P(A U B) and P(A) P(B)

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Exercise One Disjoint Events (Answers)

• Suppose we are tossing three fair coins, let the
  event A HHH, HTH, THT, and the event B
  TTT, HHT, TTH.
• Compute the size of the sample space S
• S the set of all combinations of Hs and Ts,
  so S 2 x 2 x 2 8
• Compute P(A)
• P(A) of outcomes in A / S 3/8
• Compute P(B)
• P(B) of outcomes in B / S 3/8
• Compute A U B
• A U B HHH, HTH, THT U TTT, HHT, TTH HHH,
  HTH, THT, TTT, HHT, TTH
• Compute P(A U B)
• P(A U B) of outcomes in A U B / S 6/8
• Compare P(A U B) and P(A) P(B)
• They are equal!

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Exercise Two Disjoint Events (Contd)

• Prove that if A and B are two disjoint events,
  then P(A U B) P(A) P(B)
• Proof
• P(A U B) A U B / S
• (A B) / S
• A / S B / S
• P(A) P(B)

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Exercise Three More coins tossing (Answer)

• Four coins are tossed, find the probability that
  they are not all heads.
• Let A be the event that all coins turn up a head,
  B be the event that not all coins turn up a head,
  and S be the sample space.
• B S A
• since A U B S because an outcome has to be one
  way (A) or the other (B)
• P(B) P(S A) 1 P(A) 1 1/16 15/16