Transcript Chapter 6 Section 3

Chapter 6 Section 3
Assignment of Probabilities
Sample Space and Probabilities
• Sample Space :
S = { s1 , s2 , s3 , … , sN-1 , sN }
where s1 , s2 , s3 , … , sN-1 , sN are all the possible
outcomes of the experiment
• Each outcome can be assigned a number that
represents the probability of an outcome.
Probability Distribution
Outcome
Probability
s1
p1
s2
p2
s3
p3
sN
pN
Fundamental Properties of Probability
Distributions
1.
0 < p1 < 1
0 < p2 < 1
0 < p3 < 1
0 < pN < 1
2. p1 + p2 + p3 + … + pN-1 + pN = 1
Assigning Probabilities
1. Theoretically
A. Using Logic and/or Counting Techniques
B. Best method for assigning probabilities
C. See Example 1 & 2 on page 273
2. Empirically (i.e. Experimentally)
A. Works only when the observed trial are
representative of the sample space.
B. See Example 3 on pages 273 and 274
Exercise 3 (page 281)
• Red die and green die are tossed and the number
of the side facing up are observed.
• Sample Space:
S = { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), (2,4), (2,5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }
Exercise 3 continued
• Note that each pair is equally likely to be
picked if we randomly select a pair, thus the
probability distribution is:
Outcome
Probability
s1 = (1,1)
p1 = 1/36
s2 = (1,2)
p2 = 1/36
s3 = (1,3)
p3 = 1/36
s36 = (6,6)
p36 = 1/36
Addition Principle
• Let the event E be defined as
E={s,t,u,…,z}
Then
Pr( E ) = Pr(s) + Pr(t) + Pr(u) + … + Pr(z)
Exercise 3 Part (a)
• Let E = {‘the numbers add up to 8’}
• E = { (2,6) , (3,5) , (4,4) , (5,3) , (6,2) }
• Find the Pr( E )
• Pr( E ) =
Pr( (2,6) ) + Pr( (3,5) ) + Pr( (4,4) ) +
Pr( (5,3) ) + Pr( (6,2) )
• Pr( E )
= 1/36 + 1/36 + 1/36 + 1/36 + 1/36
= 5/36
 0.1389
Inclusion – Exclusion Principle as Applied to
Probabilities
• Let E and F be events, then
Pr( E  F ) = Pr ( E ) + Pr ( F ) – Pr( E  F )
Venn Diagrams and Probabilities
We can use Venn Diagrams and place numbers that
represent the probability that the elements in the
basic region will occur.
Exercise 17 (page 282)
•
•
Let E and F be events.
Pr( E ) = 0.6 , Pr( F ) = 0.5 ,
and Pr( E  F ) = 0.4
(a) Find Pr( E  F )
(b) Find Pr( E  F)
Odds
• When the odds in favor of an event a to b ,
then the probability of the event can be
found by p = a / ( a + b )
• a represents the number of times that the
event occurs
• b represents the number of times that the
event does not occur
Converting an Odds to a Probability
• Example: If the odds of a horse to win a
race is 1 to 60, then then probability of the
horse winning is:
a = 1 and b = 60
p = a / (a + b)
= 1 / (1 + 60)
= 1/61  0.0164
Converting a Probability to an Odds
•
•
•
•
List the probability as a decimal number
Convert the decimal number to a fraction.
Reduce the fraction (when possible).
Split the value in the denominator to the
sum of two values, where the first value in
the sum is the value in the numerator.
• Get the values of a and b from the resulting
fraction.