Download Example of Sample Space 3 items are selected at random from a

Example of Sample Space
3 items are selected at random from a manufacturing process. Each item is
inspected and classified detective, D, or non detective, N.The sample space is :
={ DDD, DDN, DND, DNN, NDD, NDN, NND, NNN }
Tree Diagram of Previous Example
EVENT
Visualizing Events
1. Contingency Tables
Tree Diagrams
Complement
The complement of an event A with respect to S is the subset of all elements of
S that are not in A.
The symbol of complement of A is A’
Example :
Let R be the event that a red card is selected from
an ordinary deck of 52 playing cards, and let S be the entire deck. Then R’ is the
event that the card selected from the is not a red, but a black card
Special Events
Two events A and B are mutually exclusive, or disjoint, if A n B = Ø, that is, if A
and B have no elements in common (Events that cannot occur together)
Events A and B are mutually exclusive
Special Events
Collectively exhaustive events
One of the events must occur
The set of events covers the entire sample space
Example :
Events A, B, C and D are collectively exhaustive (but not mutually exclusive – an ace
may also be a heart)
Events B, C and D are collectively exhaustive and also mutually exclusive
Intersection & Union
Intersection of two events A and B, denoted by the symbol A n B, is the event
containing all elements that are common to A and B.
In throwing a dice, A and B are subsets of sample space S. A is the event that an
even number occurs and B the event that a number greater than 3 shows
.The union of the two events A and B, denoted by the symbol A U B, is the
event containing all elements that belong to A or B or both.
Events of Sample Space S
Some Results from Definitions
Multiplication Rule
Rule 1 : If an operation can be performed in n1 ways, and if for each of these
ways a second operation can be performed in n2 ways, the the two operations can
be performed together in n1n2 ways.
How many sample points are there in the sample space when a pair of dice is
thrown once?
A developer of a new subdivision offers prospective home buyers a choice of
Tudor, Rustic, Colonial and Traditional exterior styling in ranch, two story, and split
level floor plans. In how many different ways can a buyer order one of these homes?
Multiplication Rule
Rule 2 : If an operation can be performed in n1 ways, and if for each of these a
second
operation can be performed in n2 ways, and for each of the first two a third operation
can be performed in n3 ways, and so forth, then the sequence of k operations can be
performed in n1n2…nk ways
How many even four-digit numbers can be formed from the digits 0, 1, 2, 5, 6
and 9, if each digit can be used only once?
Permutation
A Permutation is an arrangement of all or part of a set of objects.
E.g. : Three letters a, b, and c. The possible permutations are abc, acb, bac,
bca, cab, and cba.
The number of permutations of n objects is n!
The number of permutation of n distinct objects
taken r at a time is
Permutation (cont’d)
The number of permutations of n objects arranged in a circle is (n-1)!
The number of distinct permutations of n things of which n1 are of one kind, n2
of a second kind,…, nk of a kth kind is
Example of Permutation Case
In a college football training session, the defensive coordinator needs to have 10
players standing in a row. Among these 10 players, there are 1 freshman, 2
sophomores, 4 juniors, and 3 seniors. How many different ways can they be
arranged in
a row if only their class level will be distinguished?
Answer :
Combination
Combination is selecting r objects from n without regard to order.
Example : A young boy asks his mother to get five Game-Boy cartridges from his
collection of 10 arcade and 5 sports games. How many ways are there that his
mother will get 3 arcade and 2 sports games, respectively?
Answer arcade
:
Answer Sport game
=
:
=
Probability
Probability is the numerical measure of the likelihood that an event will occur
The probability of any event must be between 0 and 1
The sum of the probabilities of all mutually exclusive and collectively exhaustive
events is 1
PROBABILITY (cont’d)
If an experiment can result in any one of N different equally likely outcomes, and if
exactly n of these outcomes correspond to event A, then the probability of event A is
Example : A statistics class for engineers consists of 25 industrial, 10 mechanical, 10
electrical, 8 civil engineering students. If a person is randomly selected by the
instructor to answer question, find the probability that the student chosen is (a)
Industrial Engineering major (b) civil engineering or electrical engineering major
Probability (cont’d)
If outcome of an experiment are not equally likely to occur, the probabilities must
be assigned on the basis of prior knowledge or experimental evidence.
For example, if a coin is not balanced, we could estimate the probabilities of
heads and tails by tossing the coin a large number times and recording the
outcomes.
According to the relative frequency definition of probability, the true probabilities
would be the fractions of heads and tails that occur in the long run
General Addition Rule
If A and B are mutually exclusive, then P(A n B) = 0, so the rule can be simplified:
General Addition Rule Example
P(Red or Ace) = P(Red) +P(Ace) - P(Red and Ace)
= 26/52 + 4/52 - 2/52 = 28/52
Additives Rules
For three events A, B, and C,
P(A U B U C) = P(A) + P(B) + P(C) – P(A n B) - P(A n C) - P(B n C) + P (A n B n C)
If A and A’ are complementary events, then
P(A) + P(A’) = 1
Example : If the probabilities that an automobile mechanic will service 3, 4, 5, 6,
7, or 8 or more cars on any given workday are respectively 0.12, 0.19, 0.28, 0.24,
0.10 and0.07, what is the probability that he will service at least 5 cars on his next
day at work?
Computing Conditional Probabilities
A conditional probability is the probability of one event, given that another event
has occurred:
Where P(A n B) = joint probability of A and B
P(A) = marginal probability of A
P(B) = marginal probability of B
P(A|B) = probability of A given B
.
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD
player (CD). 20% of the cars have both.
Conditional Probability Example
What is the probability that a car has a CD player, given that it has AC ?
i.e., we want to find P(CD | AC)
Conditional Probability Example
Of the cars on a used car lot, 70% have air conditioning (AC) and 40% have a CD
player (CD). 20% of the cars have both.
Statistical Independence
Two events are independent if and only if:
P(AIB)?P(A)
P(BIA)?P(B)
Events A and B are independent when the probability of one event is not affected by
the other event
Multiplication Rules
Multiplication rule for two events A and B:
P(A?B)?P(AIB)?P(B)
Case of Multiplication Rules
A small town has one fire engine and one ambulance available for emergencies.
The probability that the fire engine is available when needed is 0.98, and the
probability that the ambulance is available when called is0.92. In the event of an
injury resulting from a burning building, find the probability that both the ambulance
and the fire engine will be available.
Bayes’ Theorem
where:
Bi = ith event of k mutually exclusive and collectively exhaustive events
A = new event that might impact P(Bi)
Bayes’ Theorem Example
A drilling company has estimated a 40% chance of striking oil for their new well.
A detailed test has been scheduled for more information. Historically, 60% of
successful wells have had detailed tests, and 20% of unsuccessful wells have had
detailed tests.
Given that this well has been scheduled for a detailed test, what is the
probability that the well will be successful?
Bayes’ Theorem Example
Let S = successful well
U = unsuccessful well
P(S) = .4 , P(U) = .6 (prior probabilities)
Define the detailed test event as D
Conditional probabilities:
P(D|S) = .6 P(D|U) = .2
Goal is to find P(S|D)
Bayes’ Theorem Example
Apply Bayes’ Theorem:
So the revised probability of success, given that this well has been scheduled for a
detailed test, is .667
Given the detailed test, the revised probability of a successful well has risen to .667
from the original estimate of .4