Engineering Mathematics

Probability & Statistics

Mean, variance, distributions and conditional probability — recurring 1- and 2-mark scorers in every GATE CE paper.

PART 1

Topic Breakdown & Traps

The Engineering Principle

Probability quantifies uncertainty on a

[0, 1]

scale. For independent events the joint probability multiplies; for mutually exclusive events it adds. The mean locates the centre of a distribution and the variance its spread. The normal distribution is symmetric about its mean; the Poisson models rare counts.

The Core Formula Matrix

Addition:

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Independent:

P (A \cap B) = P (A) P (B)

Conditional:

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

Mean / variance (discrete):

μ = \sum x_{i} p_{i}, σ^{2} = \sum p_{i} (x_{i} - μ)^{2}

Binomial:

P (X = r) = (r n) p^{r} (1 - p)^{n - r}

The ‘IIT Traps’

⚠
Independent ≠ mutually exclusive. Independent events multiply; mutually exclusive events cannot both occur ( $P (A \cap B) = 0$ ).
⚠
Variance units are squared. Standard deviation $σ = σ^{2}$ shares the data's units.
⚠
With vs without replacement changes probabilities for the second draw.

📚 Standard references

Higher Engineering Mathematics — B.S. Grewal · Probability & Statistics
Advanced Engineering Mathematics — Erwin Kreyszig

PART 2

Progressive 3-Tier Question Suite

Q1BASIC1 Mark · NAT

A fair die is rolled once. The probability of getting a number greater than

4

is _____.

Q2MEDIUM2 Marks · NAT

Two independent events have

P (A) = 0.4

and

P (B) = 0.5

. The probability that at least one occurs is _____.

Q3HARD2 Marks · MCQ

A random variable takes values

0, 1, 2

with probabilities

0.2, 0.5, 0.3

. Its mean is: