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Class 12 Maths Notes

Probability Class 12 Notes

Complete, exam-ready notes on probability: conditional probability and the multiplication theorem, the theorem of total probability and Bayes' theorem, random variables with their probability distributions and mean-variance, and Bernoulli trials with the binomial distribution — written for CBSE boards and JEE revision.

Class12SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

Probability quantifies chance — conditional and Bayes' theorems update beliefs when new information arrives, and random variables reduce events to numbers with mean and variance.

Conditional Probability and the Multiplication Theorem

Conditional probability

P(A∣B)P(A|B)

The probability of A given that B has occurred: P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}, provided P(B)>0P(B) > 0.

  • Multiplication theorem:
  • P(A∩B)=P(A) P(B∣A)=P(B) P(A∣B)P(A \cap B) = P(A)\,P(B|A) = P(B)\,P(A|B)
  • .
  • Independent events:
  • P(A∩B)=P(A)P(B)P(A\cap B) = P(A)P(B)
  • , and P(A|B) = P(A).
  • Mutually exclusive events have
  • P(A∩B)=0P(A\cap B) = 0
  • — never confuse this with independence.

Independent vs mutually exclusive

Independent means one outcome doesn't affect the other's probability; mutually exclusive means they can't both happen. Independence gives P(A∩B) = P(A)P(B); mutual exclusivity gives P(A∩B) = 0.

Total Probability and Bayes' Theorem

Theorem of total probability

If events B₁, B₂, …, Bₙ partition the sample space, then for any event A: P(A)=∑iP(Bi) P(A∣Bi)P(A) = \sum_i P(B_i)\,P(A|B_i).

  • Bayes' theorem:
  • P(Bi∣A)=P(Bi)P(A∣Bi)∑jP(Bj)P(A∣Bj)P(B_i|A) = \frac{P(B_i)P(A|B_i)}{\sum_j P(B_j)P(A|B_j)}
  • .
  • Bayes' theorem answers 'which cause produced this result?'.
  • Common form:
  • P(A∣B)=P(B∣A)P(A)P(B)P(A|B) = \frac{P(B|A)P(A)}{P(B)}
  • .
P(A)=∑iP(Bi)P(A∣Bi),P(Bi∣A)=P(Bi)P(A∣Bi)P(A)P(A) = \sum_i P(B_i)P(A|B_i),\quad P(B_i|A) = \frac{P(B_i)P(A|B_i)}{P(A)}
Total probability and Bayes' theorem

Random Variables and Their Distribution

Random variable

A real-valued function on a sample space. Its probability distribution lists each value x with its probability p(x), which sums to 1. The mean is E(X)=∑x p(x)E(X) = \sum x\,p(x).

  • Variance:
  • Var(X)=E[(X−μ)2]=E(X2)−μ2\text{Var}(X) = E[(X-\mu)^2] = E(X^2) - \mu^2
  • .
  • Standard deviation:
  • σ=Var(X)\sigma = \sqrt{\text{Var}(X)}
  • .
  • A valid distribution requires
  • 0≤p(x)≤10 \le p(x) \le 1
  • and
  • ∑p(x)=1\sum p(x) = 1
  • .
μ=∑x p(x),Var(X)=E(X2)−μ2\mu = \sum x\,p(x),\quad \text{Var}(X) = E(X^2) - \mu^2
Mean and variance of a random variable

Bernoulli Trials and the Binomial Distribution

Binomial distribution

n independent trials, each with success probability p, give P(X=r)=(nr)pr(1−p)n−rP(X = r) = {n \choose r} p^r(1-p)^{n-r}. Mean μ=np\mu = np, variance σ2=np(1−p)\sigma^2 = np(1-p).

  • Bernoulli trials have exactly two outcomes (success/failure) and constant p.
  • The most likely number of successes is near np.
  • Sum of probabilities:
  • ∑r=0n(nr)pr(1−p)n−r=1\sum_{r=0}^n {n \choose r}p^r(1-p)^{n-r} = 1
  • .
P(X=r)=(nr)pr(1−p)n−r,μ=np, σ2=np(1−p)P(X=r) = {n \choose r}p^r(1-p)^{n-r},\quad \mu = np,\ \sigma^2 = np(1-p)
Binomial probability, mean and variance

Solved Examples

Example: A coin is tossed 3 times. Find the probability of exactly 2 heads.

Solution: This is a binomial trial with n=3n=3, p=1/2p=1/2: P(X=2)=(32)(12)2(12)1=3⋅18=38P(X=2) = {3 \choose 2}\left(\tfrac12\right)^2\left(\tfrac12\right)^1 = 3\cdot\frac18 = \frac{3}{8}.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Conditional probability

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

Multiplication theorem

P(A∩B)=P(A)P(B∣A)P(A\cap B) = P(A)P(B|A)

Independence

P(A∩B)=P(A)P(B)P(A\cap B) = P(A)P(B)

Bayes' theorem

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

Mean of random variable

μ=∑x p(x)\mu = \sum x\,p(x)

Binomial distribution

P(X=r)=(nr)pr(1−p)n−rP(X=r) = {n\choose r}p^r(1-p)^{n-r}

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • P(A|B) = P(A∩B)/P(B), defined only when P(B) > 0.
  • Independent events: P(A∩B) = P(A)P(B); mutually exclusive: P(A∩B) = 0.
  • Bayes' theorem finds P(cause | result) from P(result | cause).
  • A distribution is valid if 0 ≤ p(x) ≤ 1 and Σp(x) = 1.
  • Binomial: mean np, variance np(1−p).
  • Var(X) = E(X²) − μ².

FAQ

Common questions

What is conditional probability?

It is the probability of one event given that another has occurred: P(A|B) = P(A∩B)/P(B). It restricts the sample space to the condition B.

When are two events independent?

Two events are independent when the occurrence of one does not change the probability of the other, so P(A∩B) = P(A)P(B). This is different from being mutually exclusive.

What does Bayes' theorem do?

It reverses conditional probabilities: given the probability a hypothesis produces an observation, it computes how likely the hypothesis is after the observation — P(A|B) from P(B|A).

What is the binomial distribution?

It gives probabilities for n independent Bernoulli trials with success probability p: P(X=r) = C(n,r)·p^r·(1−p)^(n−r), with mean np and variance np(1−p).

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