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Lecture 11
Sections 5.1 – 5.2
Objectives:
•Probability
− Chance Experiments
− Sample Space and Events
− Depicting Events
− Probability Concepts
− Probability Axioms and Rules
Chance Experiments
A chance experiment (random experiment) is a procedure or an
operation whose outcome is uncertain and cannot be predicted in
advance.
To decide whether a given activity qualifies as a chance experiment,
ask yourself the question “Will I get exactly the same result if I repeat
the experiment more than once?” If answer is “no”, then the
experiment qualifies as a chance experiment.
Example
Take n=100 valves from a very large batch of valves, which contain a
proportion π of defective items.
 If several such samples are taken the number of defective items will not be the
same in each sample.
 Hypothetically suppose that we know that π =0.1. Then we would expect to get
10 defectives in the sample. A single sample may contain any number of
defectives close to "10". Keep taking samples of size 100, then we get different
defectives say 11,8, etc.
 Probability theory enables us to calculate the chance or probability a
given number of defectives.
 However in a typical experimental situation we will not know π ! Instead take a
sample of size 100 and get the number of defectives. Suppose that there were 10
defective in a sample of size 100. Then what is π ? The obvious number is 0.1.
But this varies from sample to sample. It may be 0.11 or 0.09 or any numbers
close to 0.1.
Statistical theory enables us to estimate the value π .
Probability Theory vs. Sampling Theory
In the previous example: π is a parameter.
 If the parameters of the model are known we have a probability
problem and can deduce the behavior of the system from the model.
 If the parameters are unknown and have to be estimated from the
available data then we have a statistical problem.
 The theory of probability is a good base for the study of statistics.
Basic Concepts
Sample space (denoted by S): the collection of all possible outcomes
A fair coin is tossed once: The sample space is S={H,T}, where H=Head
and T=Tail.
The lifetime of a car battery is observed.
The sample space is S=[0,∞) = {x : x ≥ 0}.
Event: a set of outcomes of a random experiment, i.e., a subset of the
sample space.
Here are some events:
o Head={H}
o Exactly one head={(H,T), (T,H)}
Forming New Events
Let A and B be two events (two subsets of sample space S),
 Union: A∪B ⊂ S (that contains all the elements that are either in A
,in B or in both).
Example. A={Sum of two dice is a multiple of 3}={3, 6, 9, 12}, B={Sum
of two dice is a multiple of 4}. Then, A∪ B ={3, 4, 6, 8, 9, 12}.
 Intersection: A∩B ⊂ S (that contains all the elements that are in
both A and B).
Example. For events A and B of the previous example, A∩ B ={12}.
 Complement of A : A' (that contains all the elements of S that are
not in A).
Example. For A of the previous example, A' ={2, 4, 5, 7, 8, 10, 11}.
 If A and B have no outcomes in common, i.e., A∩ B =ø , then A and
B are disjoint or mutually exclusive.
Example. A={Sum of two dice is even}, B={Sum of two dice is odd}.
Depicting Events
1) Venn diagrams are used to depict sample spaces, events ,
relationships among events. Consider the above example.
A∪ B
A∩ B
A‘
Mutually exclusive events
Depicting Events
2) Tree diagram is useful for depicting experiments that are
conducted in a sequence of steps in a sample space.
Example: Toss a coin three times.
Probability Concepts
Consider repeating a random experiment a number of times, say n. If
a particular outcome has occurred f times in these n trials then this
number is called the frequency of the outcome. The ratio f/n is called
the relative frequency of the outcome.
Concept of probability of event A in terms of relative frequency
A probability is a long-term relative frequency. When an experiment
is repeated a large number of times under identical conditions, a
regular pattern may emerge. In that case, the relative frequency of
event A will settle down to a fixed proportion. The fixed proportion is
defined as the probability of event A, denoted by P(A).
Coin toss
The result of any single coin toss is
random. But the result over many tosses
is predictable, as long as the trials are
independent (i.e., the outcome of a new
coin flip is not influenced by the result of
the previous flip).
The probability of
heads is 0.5 =
the proportion of
times you get
heads in many
repeated trials.
First series of tosses
Second series
Axioms
1) Probabilities range from 0
(no chance of the event) to
1 (the event has to happen).
For any event A, 0 ≤ P(A) ≤ 1
Coin Toss Example:
S = {Head, Tail}
Probability of heads = 0.5
Probability of tails = 0.5
Probability of getting a Head = 0.5
We write this as: P(Head) = 0.5
P(neither Head nor Tail) = 0
P(getting either a Head or a Tail) = 1
2) Because some outcome must occur
on every trial, the sum of the probabilities Coin toss: S = {Head, Tail}
for all possible outcomes (the sample
P(head) + P(tail) = 0.5 + 0.5 =1
space) must be exactly 1.
 P(sample space) = 1
P(S) = 1
Probability Axioms (cont d )
Venn diagrams:
A and B disjoint
3) Two events A and B are disjoint if they have
no outcomes in common and can never happen
together. The probability that A or B occurs is
then the sum of their individual probabilities.
P(A or B) = “P(A U B)” = P(A) + P(B)
This is the addition rule for disjoint events.
A and B not disjoint
Example: If you flip two coins, and the first flip does not affect the second flip:
S = {HH, HT, TH, TT}. The probability of each of these events is 1/4, or 0.25.
The probability that you obtain “only heads or only tails” is:
P(HH or TT) = P(HH) + P(TT) = 0.25 + 0.25 = 0.50
Probability Rules
The following results can be derived from axiom of probability.
Rule1. P(A') = 1− P(A) .
Rule2. P(ø ) = 0 .
Rule3. For any two events A and B, P(A∪ B) = P(A) + P(B) − P(A∩ B) .
Examples
Example: A fair coin is tossed 3 times. What is the probability getting at
least one heads?
Example: Assume that a large inventory of coated lenses includes 5
percent scratched lenses, 2 percent poorly coated, and 1 percent that
are both scratched and poorly coated. Pick randomly one lens from this
inventory. Find the probability of selecting a defective lens.