# Probability Theory solved examples & practice questions CHSL CLERK JE

#### Experiment, Outcomes, Events Probability Theory examples practice questions Experiment—A process of measurement or observations.

Randomness—A chance effect, where one cannot predict the result exactly.

Trial—Single performance of an experiment.

Outcome  (Sample  points)—Results  of an

Probability—The probability of an event A of an experiment is a measure, how frequently A is about to occur if we make many trials.

Definition 1. If the sample space of an ex- periment consists of finitely many outcomes (points), that are equally likely, then the proba- bility P(A) of an event A is

Simple event—Subsets of sample space that contain one outcome only, e.g.

An experiment is rolling a die, getting any number from 1 to 6 (uncertainty) is randomness.  1, 2, 3, 4, 5, 6 are outcomes of experiment.

S = {1, 2, 3, 4, 5, 6} is known as sample space

• {1}, {2}, … {6} are simple events
• {1, 3, 5} º  Odd  number

{2, 4, 6}  º   Even number

Getting odd number or even number is an event.

#### Union, Intersection, Complements of Events

Let S be a sample space and A, B, C, … are subsets (events) of S.

• Union A È B  =   {x : x Î A or x Î B

or x ÎA and B both}

###### (2)  Intersection

A Ç B  =   {x : x Î A and x Î B}

If A ÇB = f, then A and B are called mutually exclusive events.

###### (3)  Complement

AC   =   {x Î S and x Ï A}

• A Ç AC =  f
• A È AC =  S

and                    P(S)  =  1

PDF Probability Theory examples practice questions

Definition 2. Given a sample space S, with each event A of S (A Ì S), there is associated a number P(A), called probability of A, such that following axioms of probability are satisfied

• For every A Ì S

0  £   P(A) £ 1

• For the entire sample space

P(S)  =  1

• For mutually exclusive events A and B (A Ç B = f) [Addition rule for mutually

exclusive events] P(A È B)  =   P(A) + P(B)

• For mutually exclusive events, A1, A2,… P(A1ÈA2ÈA3 …) = P(A1) + P(A2) + …

#### Some Basic Theorems for Probability

1. Complementation rule—For an event A and its complement AC in sample space S,

P(A)  =   1 – P(AC)

1. Addition rule for mutually exclusive events—For mutually exclusive events A1,…, Am, in a sample space S,

P (A1 È A2 È  … È Am)

=  P(A1) + P(A2) + … + P(Am)

###### 3.  Addition rule for arbitrary events—

For events A and B in a sample space, P(AÈB)  =   P(A) + P(B) – P (AÇB) 