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					4.5 Probability using Tree
Diagrams & Outcome Tables
   Tree diagrams and outcome tables
    provide strategies to list outcomes of
    particular events
   Allow tracking of sequences to compute
    probability of a particular compound
    event
4.5 Probability using Tree
Diagrams & Outcome Tables
TREE DIAGRAM AND MULTIPLICATIVE PRINCIPLE
 Used to represent the outcomes of an experiment
  that are the result of a sequence of simpler
  experiments
 Assume that an outcome for each experiment has no
  influence on the outcome of any other experiment
 Total number of outcomes is the product of the
  possible outcomes at each step in the sequence.
      If a is selected from A and b from B,
                  n((a,b))=n(A) x n(B)
4.5 Probability using Tree
Diagrams & Outcome Tables
   Consider the experiment where you flip
    a fair coin and then roll a 4 sided die.
    Draw the tree diagram & list the
    outcomes.
                     1    H,1
                     2    H,2
           H
                     3    H,3
                     4    H,4
                     1     T,1
                     2     T,2
           T
                     3     T,3
                     4     T,4
4.5 Probability using Tree
Diagrams & Outcome Tables
   Consider the previous example. Would
    writing the ordered pairs as (roll, flip)
    change the effect on the total
    outcomes?
   How would the tree differ?
   What is the probability of flipping a
    head and rolling an odd number?
       P(H, odd) = 2/8 = 1/4
4.5 Probability using Tree
Diagrams & Outcome Tables
 INEPENDENT AND DEPENDENT EVENTS
In the coin toss/die roll experiment:
                   n
                   n(heads, odd rolls)
                                  rolls)
    P H odd 
    P(( H,,odd )) 
                         outcomes )
                      n(outcomes)
                     n(heads)  nodd rolls 
                                 n odd rolls
                 
                 
                   n(coin tosses) ndie rolls 
                   n(coin tosses)  n die rolls
                      nheads      nodd rolls 
                  nheads  nodd rolls 
                  ncoin tosses   ndie rolls 
                   ncoin tosses  ndie rolls 
                  P(heads)  P(odd
                  P(heads)  P(odd ))
                   1 2 1
                 12  1
                  24  4
                   2 4 4
4.5 Probability using Tree
Diagrams & Outcome Tables
   Does the outcome of the coin toss have
    any effect on the outcome of the die
    roll?
       No! They are independent events.
   Consider the conditional probability
    P(H|odd).
4.5 Probability using Tree
Diagrams & Outcome Tables
               P ( H  odd )
P( H | odd ) 
                  P (odd )
         2
        8
         1
         2

       1
     
       2

   But we know that P(H)=1/2
   Therefore, P(H|odd)=P(H) because the
    events are independent!
4.5 Probability using Tree
Diagrams & Outcome Tables
   Independent event – events A & B are said to
    be independent if the occurrence of one
    event does not change the probability of the
    occurrence of the other event
   To find the probability of several things
    happening in succession, multiply the
    probabilities of the individual happenings.
4.5 Probability using Tree
Diagrams & Outcome Tables
4.5 Probability using Tree
Diagrams & Outcome Tables
Ex.1
 A girl is told by her boyfriend that she is “one
  in a billion”. She has a dimple in her chin,
  probability 1/100, eyes of different colours,
  probability 1/1000, and is absolutely crazy
  about mathematics, probability 1/10,000.
 Are these events independent or dependent?

 Show why the girl is “one in a billion”.
4.5 Probability using Tree
Diagrams & Outcome Tables
Ex 2
 Sean Connery 007 once bet on the number 17 three
  times in succession at a roulette game in the St.
  Vincent casino. All three times the number 17 came
  up and Mr. Connery won $20,000.
       Are the events that 17 came up 3 times in a row
        independent or dependent?
   The wheel had 37 compartments, numbered 0
    through 36. What is the probability that the number
    17 would come up on:
       A single spin
       Two successive spins
       Three successive spins
4.5 Probability using Tree
Diagrams & Outcome Tables
Ex 3
 A drawer has 6 blue socks and 10 red
  socks. What is the probability of:
     Getting two blue socks
     Getting two red socks
     Getting one sock of each colour
4.5 Probability using Tree
Diagrams & Outcome Tables
Ex 3 – Solution
       Draw a tree diagram
              9/15    R
                          RR P(RR)=10/16*9/15=3/8
          R
 10/16        6/15    B
                          RB P(RB)=10/16*6/15=1/4
                                                    ¼+ ¼ = ½
              10/15   R
 6/16                     BR P(BR)=6/16*10/15=1/4
          B
              5/15    B
                          BB P(BB)=6/16*5/15=1/8
4.5 Probability using Tree
Diagrams & Outcome Tables
   Home Entertainment
       P 245 #1-4, 5ab, 6, 8, 11, 12

				
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