Transcript Chapter Two Probability

Chapter Two
Probability
Probability Definitions
Experiment:
Process that generates
observations.
Sample Space:
Set of all possible
outcomes of an
experiment.
Event Definitions
Event:
Subset of outcomes contained
in the sample space.
Simple Event:
Consists of exactly one
outcome.
Compound Event:
Consists of more than one
outcome.
Set Notation Review
For Two Events A and B:
Union: “A or B” = A  B
Intersection:
“A and B” = A  B
Complement:
A´
Mutually Exclusive:
No outcomes in common
Probabilistic Models
1) Equally Likely:
 Based on Definition
 Games of Chance
2) Relative Frequency
 Objective Interpretation
 Based on Empirical Data
3) Personal Probability
 Subjective Interpretation
 Based on Degree of
Belief
Properties of Probability
For any Event A:
P(A) = 1 – P(A)
If A and B are Mutually Exclusive,
P(A  B) = 0
For any two events A and B:
P(A  B) = P(A) + P(B) – P(A  B)
Counting Techniques
Product Rule for Ordered Pairs
Tree Diagrams
General Product Rule
Permutations
Combinations
Permutation
An “ordered” arrangement of
k distinct objects taken from a
set of n distinct objects.
The number of ways of
ordering n distinct objects
taken k at a time is Pk,n
Pk,n = n! / (n-k)!
Combination
An “unordered” arrangement
of k distinct objects taken
from a set of n distinct
objects.
The number of ways of
ordering n distinct objects
taken k at a time is Ck,n
Ck,n =
n
(
k)
= n! / k!(n-k)!
Example:
Twenty Five tickets are sold
in a lottery, with the first,
second, and third prizes to be
determined by a random
drawing. Find the number of
different ways of drawing the
three winning tickets.
Example:
Twenty tickets are sold in a
lottery, with 5 round trips to
game 1 of the World Series
to be determined by a
random drawing. Find the
number of different ways of
drawing the five winning
tickets.
Example:
A solar system contains 6
Earth-like planets & 4 Gas
Giant-like planets. How many
ways may we explore this
solar system if our resources
allow us to only probe 3 Gas
Giants and 3 Earth-like
planets?
Example:
There are 50 students in
ISE 261. What is the
probability that at least
2 students have the
same birthday? (Ignore
leap years).
Example
A dispute has risen in Watson Engineering
concerning the alleged unequal distribution of
10 computers to three different engineering
labs. The first lab (considered to be
abominable) required 4 computers; the second
lab and third lab needed 3 each. The dispute
arose over an alleged ISE 261 random
distribution of the computers to the labs
which placed all 4 of the fastest computers to
the first lab. The Dean desires to known the
number of ways of assigning the 10
computers to the three labs before deciding
on a course of action. What is the Dean’s next
question?
Conditional Probability
For any two events A and B
with P(B) > 0, the
conditional probability of A
given that B has occurred is
defined by:
P(A|B) = P(A  B)/P(B)
Multiplication Rule
P(A  B) = P(A|B) x P(B)
Multiplication Rule
Four students have responded
to a request by a blood bank.
Blood types of each student are
unknown. Blood type A+ is only
needed. Assuming one student
has this blood type; what is the
probability that at least 3
students must be typed to
obtain A+?
Conditional Probability
Experiment = One toss of a coin.
If the coin is Heads; one die is
thrown. Record Number.
If the coin is Tails; two die are
thrown. Record Sum.
What is the Probability that the
recorded number will equal 2?
Conditional Probability Problem:
30% of interstate highway accidents
involve alcohol use by at least one driver
(Event A). If alcohol is involved there is a
60% chance that excessive speed (Event S)
is also involved; otherwise, this
probability is only 10%. An accident
occurs involving speeding! What is the
probability that alcohol is involved?
P(A) = .30
P(A’)= .70
P(SA|A) = .60
P(SA’|A’)= .10
Bayes’ Theorem
A1,A2,….,Ak a collection of k
mutually exclusive and
exhaustive events with P(Ai) > 0
for i = 1,…,k. For any other event
B for which P(B) > 0:
P(Ap|B) = P (Ap  B) / P(B) =
P(B|Ap) P(Ap)
 P(B|Ai) P(Ai)
Example: Bayes’ Theorem
The probabilities are equal that
any of 3 urns A1, A2,& A3 will
be selected. Given an urn has
been selected & the drawn ball
is black; what is the probability
that the selected urn was A3?
A1 contains: 4 W & 1 Black
A2 contains: 3 W & 2 Black
A3 contains: 1 W & 4 Black
Independence
Two events A and B are
independent if:
Or
Or
P(A|B) = P(A)
P(B|A) = P(B)
P(A  B) = P(A) P(B)
and are dependent otherwise.
Independence Example:
Three brands of coffee, X, Y,& Z are to
be ranked according to taste by a
judge. Define the following events as:
A: Brand X is preferred to Y
B: Brand X is ranked Best
C: Brand X is ranked Second
D: Brand X is ranked Third
If the judge actually has no taste
preference & thus randomly assigns
ranks to the brands, is event A
independent of events B, C, & D?
Independence
Consider the following 3
events in the toss of a single
die:
A: Observe an odd number
B: Observe an even number
C: Observe an 1 or 2
Are A & B independent events?
Are A & C independent events?
Example:
A space probe to Mars has 35
electrical components in
series. If the mission is to
have a reliability (probability
of success) of 0.90 & if all
parts have the same
reliability, what is the
required reliability of each
part?