Paper I — Mathematics

Probability

Exam-focused notes, visual explanation, traps, revision sheet and every mapped past NDA question.

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High yield5 mapped NDA questions

Build the idea

Probability is one of the highest-return NDA chapters because it combines counting with a small set of laws. Define the sample space carefully, test independence separately from mutual exclusivity, and use complements for 'at least one' questions.

Union, intersection and overlap

Concept

Probability measures the chance of an event, from 0 (impossible) to 1 (certain). The classical definition is favourable outcomes over total outcomes. Conditional probability, Bayes' theorem and the binomial distribution build on this.

Identify whether events are independent or mutually exclusive before applying rules.

Key Points & Formulas

P(E) = favourable / total; 0 ≤ P(E) ≤ 1.
Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B).
Independent events: P(A∩B) = P(A)·P(B).
Binomial: P(r successes) = nCr pʳ q^(n−r); mean = np.

Worked Example

Q. Two fair dice are thrown. Find the probability that the sum is 9.

Favourable pairs: (3,6),(4,5),(5,4),(6,3) = 4 outcomes.

Total outcomes = 36.

Probability = 4/36 = 1/9.

Common traps

Treating mutually exclusive events as independent.
Adding probabilities without subtracting the intersection.
Assuming outcomes are equally likely without justification.

NDA shortcuts

P(at least one)=1−P(none).
For independent trials, multiply along a path and add alternative paths.
Use combinations when order does not matter.

One-screen revision sheet

Recall before you practise

P(A∪B)=P(A)+P(B)−P(A∩B)
P(A|B)=P(A∩B)/P(B)
Independent: P(A∩B)=P(A)P(B)

Past-paper bank

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5 NDA questions

Choose an answer before the solution appears. Option order is mixed automatically.

Question 1

NDA I 2026

In an entrance test there are multiple choice questions. There are four options for each question, of which only one is correct. The probability that a student knows the answer to a question is $90\%$ . If he gets the correct answer to a question, then what is the probability that he was guessing?

Question 2

NDA II 2025

Let $X$ be a random variable following binomial distribution with parameters $n = 5$ and $p = k$ . Further, $P(X = 1) = 0.4096$ and $P(X = 2) = 0.2048$ . What is the value of $k$ ?

Question 3

NDA II 2024

Urn $A$ contains $2$ white and $2$ black balls while urn $B$ contains $3$ white and $2$ black balls. One ball is transferred from urn $A$ to urn $B$ and then a ball is drawn out of urn $B$ . What is the probability that the ball is white?

Question 4

NDA II 2024

A die has two faces with number $4$ , three faces with number $5$ and one face with number $6$ . If the die is rolled once, then what is the probability of getting $4$ or $5$ ?

Question 5

NDA I 2024

What is the value of p + q?

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