# Slide 1 - Thames Valley District School Board

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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
rolls)
P H odd 
P(( H,,odd )) 
outcomes )
n(outcomes)
n odd rolls


n(coin tosses) ndie rolls 
n(coin tosses)  n die rolls
 nheads  nodd rolls 
 ncoin tosses   ndie rolls 
ncoin tosses  ndie rolls 
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
 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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