mal eta adep summa m eta alat settab222.ppt

Probability
 Definition
 Concepts in probability
 Methods of Assigning probability

Experiment
• An experiment is the process of making an
observation

Random Experiment
The actual outcome in a Random
Experiment cannot be determined
in advance. i.e if the experiment
is repeated the outcome may be
different

Events
Outcomes of random experiments are called events
Suppose we perform an Random experiment
where we toss a coin and observe whether the
upside of the coin is a head or a tail
There are two possible outcomes to the
experiment i.e upside of the coin may be a
head or tail

Mutually Exclusive Events
Two events are said to be mutually exclusive if,
when one of the events occurs the other event
cannot occur
EG When tossing a coin, the outcome can
only be heads (H) or tails (T), not heads and
tails, therefore the events H and T are
mutually exclusive

Events
The complement of an event A, denoted by A, is
the set of outcomes that are not in A. A means A
does not occur
The union of two events A and B, denoted by A U
B, is the set of outcomes that are in A, or B, or both
The intersection of two events A and B, denoted by
AB, is the set of outcomes that are in both A and B

Sample Space
A universe of elementary outcomes.

Random Variable

Random variable is used to express
Outcomes (Events) of an experiment
Numerically

Random Variable
e.g.:
a) Toss a Coin;
b) observe for head(this is the event).
c) Assign 0 if Tail comes and 1 if head comes
to a variable X
Hence X is A random variable used to denote
The event head(Happening of head)

Random Variable
e.g.:
a) Toss a Dice two times;
b) observe for number 4(this is the event).
c) Assign 0,1,2 to a variable X
Hence X is A random variable used to denote
The event 4 (Getting 4)

Discrete Random Variable
 Discrete random variable
 Obtained by counting (1, 2, 3, etc.)
 Usually a finite number of
different values
 e.g.: Toss a coin five times;
Count the number of tails
(0, 1, 2, 3, 4, or 5 times)

Discrete Random Variable
Examples
Experiment Random
Variable
Possible
Values
 DO 100 Surgery # Alive 0, 1, 2, ..., 100
 Diagnose 70 Patients # Affected 0, 1, 2, ..., 70
 Answer 33 Questions # Correct 0, 1, 2, ..., 33
 Count stars at night # stars 0, 1,
2, ...,between 0 & ∞ at night

Probability Definition
• A phenomenon is called random if the outcome of
an experiment is uncertain
• Random phenomena often follow recognizable
patterns.
• This long-run regularities of random phenomena
can be described Mathematically.
• The mathematical study of randomness is called
probability theory – Probability provides a
mathematical description of randomness.
• Probability is a Chance of happening of event.
• It measures likelihood of an event

Methods of Assigning
Probability
 Classical Probability
 Relative Frequency Probability
 Subjective Probability

Classical Probability
Calculating Probability when the
possible outcomes for an event can
be mathematically derived. Usually
the events are equally likely.
EG A Fair Coin P(heads) = 0.5
A Dice P(1) = 0.1667
A Deck of Cards P(club) = 0.25

Classical Probability
 P(A) = Favorable Number of cases for Event
A
Total Number of cases

Relative frequency probability
probability based on observations from a
large number of trials or historical
records.
Records show that it has rained in
Dharan on 13 Sundays of the last 52
Sundays, therefore
P(rain in Dharan on a Sunday) = 13/52 =
0.25
.

Relative Frequency Probability
 P(A) = Frequency of Event A
Total Frequency

Subjective probability
An educated guess! When there are no
precise mathematics and no large number
of historical trials available
EG When you wake up in the morning,
look out the window and figure that
because there are no clouds it won’t rain
today, so don’t take your umbrella with
you

Properties of probability
The probability of any event is always
between 0 and 1
If we list all possible mutually exclusive
events associated with an experiment,
then the sum of their probabilities will
always equal 1

Conditional probability
The probability of an event A, given that B
has occurred, is called the conditional
probability of A given B and is denoted by
P(A|B)
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A survey has the following data on the
age and marital status of 140 Patients
MARITAL STATUS
AGE Single (S) Married (M) TOTAL
<30 (L) 77 14 91
>30 (S) 28 21 49
Total 105 35 140

If a Patient is under 30, what is the
probability that he is single?
i.e find P(S|L)

Ans…
There are 91 Patients under 30, and 77 of
those are single. Therefore;
P(S|L) = 77/91 = 0.846

Independent Events
Two events A and B are said to be
independent if;
(i)P(A|B) = P(A), or
(ii)P(B|A) = P(B)
(iii)P(A n B) = P(A) x P(B)

Independent Events
Example; Consider the following data;
Disease Male
Male Female
Female total
total
Affected
Affected 23
23 92
92 115
115
Not Affected
Not Affected 4
4 16
16 20
20
total
total 27
27 108
108 135
135
We want to test whether Disease Status is independent of Gender

Independent Events
P(Male) = 27/135 = 0.2
P(Male|Affected) = 23/115 = 0.2
Therefore, as P(Male) = P(Male|Affected),
Disease Status is independent of gender

Laws of Probability
.Addition Law
.Addition Law
.Multiplication Law
.Multiplication Law

Addition Law
Addition Law
The Addition law provides a way to compute
the probability of event A or B or both A and B
occurring.
The law is written as:
P(A u B) = P(A) + P(B)

Multiplication Law
Multiplication Law
Multiplication Law for Independent Events:
P(A n B) = P(A)P(B)
The multiplication law also can be used as a test
to see if two events are independent.

Bayesian Law
Bayesian Law
AB
~AB
B
B
A
A

Bayesian Law
Bayesian Law
One patient was diagnosed positive for
tuberculosis.He told that he won't smoke.
Calculate the probability that the patient smoked
P(Smoking) = 0.2
P(Not smoking) = 0.8
P(Smoking and Positive for TB) = 0.9
P(Smoking and not positive for TB) = 0.1
P(Not Smoking and Positive for TB) = 0.1
P(Not Smoking and not positive for TB) = 0.9

Bayesian Law
Bayesian Law
F-Smoking
E-Positive for TB
P(F n E)=0.18
P(F n E’)=0.02
P(F’ n E) = 0.08
P(F’ n E’) = 0.72
P(E’/F) = 0.1
P(E/F)=0.9
P(E/F’) = 0.1
P(E’/F’) = 0.9

Bayesian Law
Bayesian Law
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mal eta adep summa m eta alat settab222.ppt

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