Transcript Introduction to Decision Analysis

Counting
Techniques
(Dr. Monticino)
Overview
Why counting?
Counting techniques
Multiplication principle
Permutation
Combination
Examples
Probability examples
Why Counting?
 Recall that if each outcome of an experiment
is assumed to be equally likely, then the
probability of an event is k/n
 where k is the number of elements in the event
and n is the number of elements in the sample
space
 So to calculate the probability of an event, we
need to be able to count the number of
elements in the event and in the sample space
Multiplication Principle
Multiplication principle. Suppose that an
experiment can be regarded as a series of k subexperiments. Such that the first sub-experiment
has n1 possible outcomes, the second subexperiment has n2 possible outcomes, and so on.
Then the total number of outcomes in the main
experiment is
 n1 x n2 x ... x nk
Examples
 Flip a coin and roll a die
 Roll 5 die; or roll a single die five times
Permutation
Factorial. n! (read “n factorial”) equals
n  (n  1)  (n  2) 2  1
Permutation. The number of ways to
select r objects, in order, out of n objects
equals
n!
n  (n  1)  (n  2) (n  r  1) 
(n  r )!
Examples
How many ways are there to do the
following
Line up 10 people
Select a President, VP and Treasurer from a
group of 10 people
Sit 5 men and 5 women in a row,
alternating gender
Combination
Combination. The number of ways to
select r objects out of n objects when order
is not relevant equals
 n
n!
 
 r  r !(n  r )!
Examples
How many ways are there to do the
following
Select 3 people from a group of 10
Select 7 people from a group of 10
Get exactly 5 heads out of 12 coin flips
Probability Examples
Select three people at random from a
group of 5 women and and 5 men
What is the probability that all those
selected are men?
What is the probability that at least one
women is chosen?
What is the probability that at least two
women are chosen?
Probability Examples
Flip a fair coin 3 times
What is the probability that 3 heads come
up?
What is the probability that at least 1 tail
occurs?
What is the probability that exactly 2 tails
occur?
What is the probability that at least 2 tails
occur?
Probability Examples
Play roulette 3 times
What is the probability that red comes up
every time?
What is the probability that black comes up
at least once?
What is the probability that black comes up
exactly two times?
What is the probability that black comes up
at least two times?
Probability Examples
Flip a fair coin 10 times
What is the probability that 10 heads come
up?
What is the probability that at least 1 tail
occurs?
What is the probability that exactly 8 tails
occur?
What is the probability that at least 8 tails
occur?
Probability Examples
Play roulette 20 times
What is the probability that red comes up
every time?
What is the probability that black comes up
at least once?
What is the probability that black comes up
exactly 18 times?
What is the probability that black comes up
at least 18 times?
Probability Examples
Roll a fair die 5 times
What is the probability that an ace comes
up all five times?
What is the probability that an ace occurs
at least once?
What is the probability that an ace occurs
exactly 3 times?
What is the probability that an ace occurs
at least 3 times?
Probability Examples
 To win the jackpot in Lotto Texas you need to
match all six of the numbers drawn (5
numbers are selected from numbers 1 to 44
and the sixth is selected separately from 1 to
44)
 What is the probability of winning if you buy one
ticket?
 What is the probability of winning if you buy five
tickets?
 Is it better to buy five tickets in one Lotto drawing
or a single ticket in five successive Lotto games?
Assignment Sheet
Read Chapter 15 carefully
Redo all problems from lecture
Not to turn in…
(Dr. Monticino)