Model Institute of Engineering & Technology (Autonomous)

Department of Computer Science Engineering

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Course Name- Engineering Mathematics

Course Code - BSC-201

Topic- Conditional Probability

Faculty- Neha Malhotra

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Course Outcomes

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Learning Outcomes

• Articulate the concept of Probability, Event and Sample Space of an experiment.
• Solve the different problems of Probability
• Apply the concept of probability in day-to-day life

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Two Worlds

DETERMINISTIC IBM Plex Mono

PV = C

Predictable Outcome. Result is unique.

PROBABI LISTIC

IBM Plex Mono

P(A) + P (B) = P(AUB)

Predictable Chance. Result is unknown.

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Mapping the Territory

S

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Map Legend

Union

(A U B)

Intersection

(AUB)

Complement

(A^C)

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The Rules of Arrangement

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PERNUTATION

Order Matters.

nP

n!

” " (n-z) !

Arrangement.

CONBINATION

Order Does Not Matter.

nC, = ^n!

(n-r)! r!

Selection.

The Classical Definition

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ò lmpossibility

Total cases

P(E)

Favorable cases

Certainty

Relationships in Space

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Collectively Exhaustive

Union equals Sample Space

Complementary

P(A) + P(A^c) = 1

The Addition Theorem

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Subtract the overlap

P(A U B)

Conditional Probability

Given that A has occurred...

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The sample space shrinks. We define the probability of B relative to the area of A.

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The Multiplication Theorem

Calculating simultaneous occurrenee.

Event A

P(A)

P(A

AND THEN

Event B

P(B |A)

P(A) • P(B|A)

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Independence

Two events are independent

if the occurrence of one does

not affect the other.

P(B |A) = P(B)

Event A: Coin 1 is Heads

— — —X— — - X

NO CONNECTION

Event B: Coin 2 is Tails

Cr‹ution: Do not confuse with Mutually Exclusive. Independent events can happen together; they just don't influence each other.

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Theorem of Total Probability

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P(A)

Bayes’ Theorem

Inverse Probability

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The Specific Path

(E0ect from Specific Cause)

^P(B_i)•P(A|B_i)

P(Bj)

.P(A|Bj))

Total Probability

(Sum of All Paths)

Reasoning from the observed

Effect (A) back to the probable Cause (B).

Case Study: The Defective Car

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Tot:a1 Product:1on

Factory B,

3õîb 3õä

45%

45%

Factory B₂

25îb

25Ø

Factory B₃

2%

3%

2%

Problem: We have a defective car (A). What is the probability

it came from Factory B₂?

Solving for the Source

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Step 1. Tota! Probability (Denominator)

(0.30) (0. 02) + (0. 45) (0. 03) + (0.25) (0. 02)

0. 0245

Step 2. Target Path Factory B₂ (Numerator)

(0. 45) (0. 03) = 0. 0235

Step 3. Bayes’ Calculation

P(B₂|A) = â. â135

â. â245 -

â. 55

55% Probability

Conditional Probability

If A and B are two events A and B such that P(A) occurs only when P(B)

has already take place, the conditional probability of A given B is.

Reflective Questions

• We flip two coins. What is the probability that both are heads, given that at least one of them is heads .
• We roll a six-sided die. What is the probability that the roll is a 6, given that the outcome is an even number
• Given that E and F are two events such that P(E)=0.6, P(E∩F)=0.2

P(F)=0.3. Calculate P(E|F) and P(F|E).

• Determine P(E|F) in all the cases

(a) E: atmost 2 heads, F: atleast 2 heads. When three coins are tossed simultaneously.

• A dice is thrown three times, E: 4 appears on the third toss, F: 6 and 5 appears on the first two toss.

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Formulae at a Glance

(Ozdez matters)

(Selection

only)

A B

Mathematics is the logic of certainty applied to the world of chance.

Summarize

• Conditional probability measures the likelihood of an event occurring given that another event has already happened

Thank You

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