Probability

part 1

Conditional Probability

• Conditional probability — definition and properties
• Multiplication rule of probability
• Independent events
• Properties of conditional probability

part 2

Total Probability and Bayes' Theorem

• Law of total probability
• Bayes' theorem
• Applications of Bayes' theorem
• Partition of sample space

part 3

Random Variables and Distributions

• Random variables — discrete and continuous
• Probability distribution of a random variable
• Mean and variance of a random variable
• Bernoulli trials and Binomial distribution
• Mean and variance of binomial distribution

Conditional probability

$P(A|B) = \frac{P(A\cap B)}{P(B)},\ P(B) \neq 0$

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Multiplication theorem

$P(A\cap B) = P(A)\cdot P(B|A) = P(B)\cdot P(A|B)$

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Independent events

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

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Total Probability theorem

$P(A) = \sum_i P(E_i)\cdot P(A|E_i)$

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Bayes' theorem

$P(E_i|A) = \frac{P(E_i)P(A|E_i)}{\sum_j P(E_j)P(A|E_j)}$

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Binomial distribution P(X = r)

$\binom{n}{r} p^r q^{n-r}$ (q = 1−p)

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Mean of Binomial distribution

$\mu = E(X) = np$

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Variance of Binomial distribution

$\sigma^2 = npq$

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Expected value E(X)

$E(X) = \sum x_i \cdot P(x_i)$

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Variance Var(X)

$\text{Var}(X) = E(X^2) - [E(X)]^2 = \sum x_i^2 P(x_i) - [E(X)]^2$

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Of the students in a college, 60% reside in a hostel and 40% do not. Previous results show that 30% of hostellers and 20% of non-hostellers attain an A grade. A student chosen at random has an A grade. Find the probability that the student is a hosteller.

A and B are two independent events such that P(A) = 1/2 and P(B) = 1/3. Find P(A ∪ B), P(A ∩ B) and P(A|B).

A factory has machines A, B, C producing 50%, 30%, 20% of output. The defective rates are 1%, 2%, 3% respectively. A defective item is found. Find the probability it came from machine B using Bayes' theorem.

Two cards are drawn simultaneously (without replacement) from a well-shuffled pack of 52 cards. Find the probability distribution of the number of kings and hence its mean.

A fair coin is tossed 8 times. Find the probability of getting (a) exactly 5 heads, (b) at least 6 heads.

There are three coins. One is a two-headed coin, another is a biased coin that comes up heads 75% of the time, and the third is unbiased. One coin is chosen at random and tossed, showing heads. What is the probability that it was the two-headed coin?

A card is drawn from a deck of 52 cards. Find the probability that it is a king given that it is a face card.

The probability that a bulb produced by a factory is defective is 1/50. A sample of 5 bulbs is drawn at random. Find the probability that (a) none is defective, (b) exactly one is defective.

A die is thrown twice and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared at least once?

Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X and its mean.

If P(A) = 0.3, P(B) = 0.6 and A and B are independent events, find P(A ∩ B).

How many marks is Probability worth in the Class 12 board exam?

Around 8 marks of the 80-mark CBSE Maths theory paper, based on the official unit-wise weightage. Bayes' theorem and binomial distribution — most asked.

Is Probability easy or hard?

It's rated hard — one of the tougher chapters in Class 12 Maths. Most students need about 4 hours to cover it well.

What are the important topics in Probability?

The chapter covers 12 NCERT topics in 3 areas: Conditional Probability; Total Probability and Bayes' Theorem; Random Variables and Distributions.

What questions come from Probability in board exams?

Between 2018–2024, CBSE board papers asked questions from this chapter on Bayes' Theorem, Independent Events, Probability Distribution, Binomial Distribution — 1- to 5-mark questions.