Submitted By Shubham2609

Words 1240

Pages 5

Words 1240

Pages 5

Sessions 1-5

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Basic Concepts-I

• An experiment is an occurrence whose result, or outcome,

is uncertain.

• The set of all possible outcomes is called the sample space for the experiment.

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Example

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One Die

Two Dice

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Basic Concepts-II

• Given a sample space S, an event E is a subset of S.

• The outcomes in E are called the favourable outcomes.

• We say that E occurs in a particular experiment if the

outcome of that experiment is one of the elements of E.

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Examples

• When a fair die is rolled, probability of each outcome is 1/6.

• So probability of E: Even numbers is P(E) = 3/6 = 0.5.

• But a die does not need to be fair, and outcomes are not

always equally likely!

• For example, suppose we have a loaded die with probabilities of 1, 2, 3, 4, 5 and 6 resp. 0.25, 0.15, 0.15, 0.15,

0.15 and 0.15.

• Then P(E) = 0.45.

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Basic Concepts

For any event A:

i) 0 ≤ P(A) ≤ 1. ii) P(Φ) = 0, where Φ is the null event (no outcomes). iii) P(S) = 1, where S is the sample space (all outcomes). iv) If A and B are two events with no common outcome, (i.e.

“mutually exclusive”) then

P(A B) P(A) P(B)

v) If A and B are two mutually exclusive events such that

P(A B) 1 then A and B are called complements of each other. We denote B as A or AC.

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General Formula

P(A B) P(A) P( B) P(A B)

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Example

• Consider the fair die being rolled twice

• P(sum is even) =

• P(product is odd) =

• P(sum is even, product is odd) =

• Re-compute the probabilities for the loaded die.

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Example

Work out the probabilities

i) P(sum is even given the product is odd) ii) P(product is odd given sum is even) for the fair die as well as the loaded die.

How to proceed?

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Conditional Probability

• Let A, B be two events such that P(A) > 0. Then

P ( A B)

P( B | A) …...