Probability

1. -By Ishara Saranapala

2.  Definition of Probability.
 Probability rules.
 Probability approaches.
 Conditional Probability.

3.  Probability is a branch of mathematics and it explains the ‘chance’
that something will happen.
 In other words, Probability is the measure of the likeliness that an
event will occur. (wikipedia)
That deals with calculating the likelihood of a given event's
occurrence, which is expressed as a number between ‘1’ and ‘0’.
Ex: The toss of a fair coin. Since the two outcomes are equally
probable, the probability of "heads" is 0.5 and probability of "tails", is
0.5.

4. Collecting Data
Exploring Data
Probability Intro.
Inference
Comparing Variables Relationships between Variables
Means Proportions Regression Contingency Tables

5.  We have several graphical and numerical
statistics for summarizing our data
 To make probability statements about the
significance of the statistics
 Ex: Mean(income) = Rs.55,000
◦ What is the chance that the true income of Employees of
‘X’ company is between Rs. 50,000 and Rs.70,000?
 For example; r = 0.82 for educational qualification
and salary level
◦ What is the chance that the true correlation is
significantly different from zero?

6.  In deterministic processes, the outcome can be
predicted exactly in advance
 In random processes, the outcome is not known
exactly, but we can still describe the probability
distribution of possible outcomes

7.  An event is an outcome or a set of outcomes of
a random process
◦ Example: Tossing a coin three times
◦ Example: Tossing a fair dice
 * Notation: P(A) = Probability of event A

8.  Rule 1: 0 ≤ P(A) ≤ 1 for any event A
-The sample space S of a random process is the set of
all possible outcomes
 Probability Rule 2: The probability of the
whole sample space is 1 P(S) = 1
-The complement Ac
of an event A is the event that A
does not occur

9.  Probability Rule 3:
P(Ac
) = 1 - P(A)
 The union of two events A and B is the event that either A or B
or both occurs
 The intersection of two events A and B is the event that both A
and B occur
-Disjoint Events:
 Two events are called disjoint if they can not happen at the same time.
 Ex: coin is tossed twice
S = {HH,TH,HT,TT}
Events A= {HH} and B= {TT} are disjoint .
(Events A and B are disjoint means that the intersection of A and B is
zero)

10.  Probability Rule 4: If A and B are disjoint
events then : P(A or B) = P(A) + P(B)
Events A and B are independent if knowing that A occurs
does not affect the probability that B occurs.
Ex: Tossing two coins
Event A = first coin is a head
Event B = second coin is a head
- ‘Disjoint events cannot be independent!’ - If A and B can not
occur together (disjoint), then knowing that A occurs does change probability
that B occurs

11.  Probability Rule 5: If A and B are independent
P(A and B) = P(A) x P(B)
-Mutually Exclusive Events:
Two or more events are said to be mutually exclusive if at most
one of them can occur when the experiment is performed, that is,
if no two of them have outcomes in common.
(a) Two mutually exclusive events
(b) Two non-mutually exclusive events

12.  If all possible outcomes from a random process
have the same probability, then
◦ P(A) = (# of outcomes in A)/(# of outcomes in S)
 Ex: One Dice Tossed
P(even number) = |2,4,6| / |1,2,3,4,5,6|
 Note: equal outcomes rule only works if the
number of outcomes is “countable / finite”

13.  If, S (ample space) = {O1, O2, …, Ok}, the
probabilities assigned to the outcome must
satisfy these requirements:
(1) The probability of any outcome is between 0 and 1
i.e. 0 ≤ P(Oi) ≤ 1
(2) The sum of the probabilities of all the outcomes
equals 1
P(Oi) represents the probability of outcome i

14. There are three ways to assign a probability, P(Oi), to
an outcome, Oi, namely:
Classical approach: Make certain assumptions
(such as equally likely, independence) about situation.
Relative frequency: Assigning probabilities based
on experimentation or historical data.
Subjective approach: Assigning probabilities
based on the assignor’s judgment. [Bayesian]

15. If an experiment has n possible outcomes [all
equally likely to occur], this method would assign
a probability of 1/n to each outcome.
 Experiment: Rolling a dice
◦Sample Space: S = {1, 2, 3, 4, 5, 6}
◦Probabilities: Each sample point has a 1/6
chance of occurring.

16. Experiment: Rolling 2 dice and summing 2 numbers
on top.
Sample Space: S = {2, 3, …, 12}
P(2) =
1/ 36
P(6) =
5/ 36
P(10) =
3 / 36
1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12

17. Bits & Bytes Computer Shop tracks the number of
desktop computer systems it sells over a month (30
days):
For example,
◦10 days out of 30
◦2 desktops were sold.
From this we can construct
◦the “estimated” probabilities of an event
◦(i.e. the # of desktop sold on a given day)…
Desktops Sold # of Days
0 1
1 2
2 10
3 12
4 5

18. “In the subjective approach we define probability as
the degree of belief that we hold in the occurrence of
an event”
Ex:
P (You drop from the country)
P (NASA successfully land a man on the Mars)

19.  Let A and B be two events in sample space
The conditional probability that event B
occurs given that event A has occurred is:
P(A|B) = P(A and B) / P(B)

20.  The concept of conditional probability can be found
in many different types of problems
 Ex: Smoking and Lung Cancers among 60 to 65 year old
men
 The probability that a person is a smoker is.
40/100 = 0.4
What is the probability that a person is a smoker given that they are
suffering from lung cancer?
 30 / 40 = 0.75
Smoker Non-smoker Total
Lung Cancer 30 10 40
No Lung Cancer 10 50 60
Total 40 60 100