# 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

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

P(A) = 1 – P(AC)

**Addition rule for mutually exclusive events—**For mutually exclusive events A1,…, A*m*, in a sample space S,

P (A1 È A2 È … È A*m*)

= P(A1) + P(A2) + … + P(A*m*)

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

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

# Probability Theory solved examples & practice questions

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