Tp4 probability

-By Ishara Saranapala

Definition of Probability.

Probability rules.

Probability approaches.

Conditional Probability.

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.

Collecting Data

Exploring DataProbability Intro.

Inference

Comparing Variables Relationships between Variables

Means Proportions Regression Contingency Tables

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?

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

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

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

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)

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

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

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”

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 1i.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

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]

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.

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

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

“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)

Let A and B be two events in sample spaceThe conditional probability that event B occurs given that event A has occurred is:

P(A|B) = P(A and B) / P(B)

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