Probability — Study Notes

1. Introduction

Probability is the mathematics of chance, quantifying how likely events are to occur. Class 11 introduced the classical definition of probability for equally likely outcomes. This chapter generalises the theory to conditional probability, where the likelihood of one event is recalculated given that another event has already occurred, and builds the machinery of the multiplication rule, the law of total probability, and Bayes' theorem.

The chapter then introduces random variables and their probability distributions, the concept of the mean and variance of a distribution, and concludes with the binomial distribution, which models the number of successes in a fixed number of independent trials. These tools are indispensable for statistics, genetics, quality control, decision theory, and game theory.

In the board examination, this chapter typically carries one of the highest weightages, with both short objective questions and long-answer problems involving Bayes' theorem and binomial distributions. The concepts are subtle, and the errors are characteristically conceptual rather than computational: mislabelling conditional probabilities, swapping P(A|B) and P(B|A), and misidentifying the parameters of the binomial distribution. Careful reading of the problem statement is half the battle in this chapter.

2. Conditional Probability

If E and F are events associated with a random experiment, the conditional probability of E given F, written P(E|F), is the probability that E occurs given that F has already occurred:

$$P(E|F) = \frac{P(E \cap F)}{P(F)}, \quad P(F) > 0$$

Properties

• $P(E|F) \in [0, 1]$
• $P(E'|F) = 1 - P(E|F)$
• $P((E_1 \cup E_2)|F) = P(E_1|F) + P(E_2|F) - P((E_1 \cap E_2)|F)$

3. Multiplication Theorem and Independent Events

Multiplication Theorem

$$P(E \cap F) = P(E) \cdot P(F|E) = P(F) \cdot P(E|F)$$

For three events, $P(E \cap F \cap G) = P(E) \cdot P(F|E) \cdot P(G|E \cap F)$.

Independent Events

Two events E and F are independent if

$$P(E \cap F) = P(E) \cdot P(F)$$

which is equivalent to $P(E|F) = P(E)$ (when P(F) > 0): knowing F does not change the probability of E. Independent events satisfy the important closure properties:

$$P(E \cap F') = P(E) - P(E \cap F) = P(E)P(F')$$

• If E and F are independent, so are E and F', E' and F, and E' and F'.
• Independent is NOT the same as mutually exclusive; mutually exclusive events with positive probability are always dependent (since both cannot occur, P(E and F) = 0 while P(E)P(F) > 0).

4. Partition of a Sample Space and the Law of Total Probability

Events $E_1, E_2, \ldots, E_n$ form a partition of the sample space S if they are pairwise disjoint and their union is S. Then for any event A:

$$P(A) = \sum_{i=1}^{n} P(E_i) \cdot P(A|E_i)$$

This is the law of total probability: the probability of A is the weighted average of the conditional probabilities over the partition.

5. Bayes' Theorem

Given a partition $E_1, \ldots, E_n$ of the sample space and an event A with P(A) > 0:

$$P(E_i|A) = \frac{P(E_i) \cdot P(A|E_i)}{\sum_{j=1}^{n} P(E_j) \cdot P(A|E_j)}$$

The probabilities $P(E_i)$ are called the prior probabilities, and $P(E_i|A)$ the posterior probabilities. Bayes' theorem updates our belief about which event of the partition actually occurred after observing the evidence A.

6. Random Variables and Probability Distributions

A random variable X is a real-valued function defined on the sample space. It can be discrete (taking finitely or countably many values) or continuous. This chapter focuses on discrete random variables.

The probability distribution of a discrete random variable lists each value $x_i$ with its probability $p_i = P(X = x_i)$, where $p_i \geq 0$ and $\sum p_i = 1$.

Mean (Expectation)

$$E(X) = \mu = \sum x_i p_i$$

Variance and Standard Deviation

$$Var(X) = \sigma^2 = \sum (x_i - \mu)^2 p_i = \sum x_i^2 p_i - \mu^2$$

$$\sigma = \sqrt{Var(X)}$$

7. Binomial Distribution

A binomial experiment consists of n independent trials, each with two outcomes (success/failure), where the probability of success p is the same in every trial. If X is the number of successes, then X follows the binomial distribution with parameters n and p:

$$P(X = r) = \binom{n}{r} p^r q^{n-r}, \quad r = 0, 1, \ldots, n$$

where $q = 1 - p$.

Mean and Variance of a Binomial Distribution

$$\mu = np, \qquad \sigma^2 = npq$$

The conditions for a binomial experiment: fixed number of trials n, each trial independent, each trial has exactly two outcomes, and p is constant.

Quick Revision Tables

Table 1: Key Formulas

Table 2: Independent vs Mutually Exclusive

Table 3: Binomial Distribution Properties

Mind Map

Important Diagrams (SVG)

Diagram 1: Tree Diagram for Bayes' Theorem

Diagram 2: Binomial Distribution for n = 5

Common Mistakes

1. Confusing P(E|F) with P(F|E). These are generally different; Bayes' theorem is needed to convert between them.
2. Treating mutually exclusive events as independent. If E and F are mutually exclusive with positive probability, then P(E and F) = 0 but P(E)P(F) > 0, so they are dependent.
3. Forgetting that P(E|F) is defined only when P(F) > 0.
4. In the law of total probability, failing to ensure the events Ei form a partition of the sample space.
5. In Bayes' theorem, using the wrong event as the denominator; the denominator must sum over all prior events.
6. For a random variable, forgetting that the probabilities must sum to 1; verify this after writing the distribution.
7. Using the formula E(X^2) - (E(X))^2 with the wrong pairing; variance is E(X^2) - (E(X))^2, where E(X^2) = sum xi^2 pi.
8. Misidentifying p and q in the binomial distribution; q = 1 - p is the failure probability, and P(X = r) uses p^r q^(n-r).
9. Using npq for the standard deviation; the variance is npq and the standard deviation is sqrt(npq).
10. Assuming independence when trials are not independent (e.g., drawing cards without replacement), which invalidates the binomial model.

Exam Tips

1. Read the problem and label the events clearly before writing any formula; decide what P(E|F) versus P(F|E) means in the context.
2. For Bayes' theorem problems, draw a tree diagram showing the partition and the evidence event; this makes the numerator and denominator unambiguous.
3. To test independence, compare P(E and F) with P(E)P(F); equality means independence.
4. For the law of total probability, list the partition events explicitly and compute each P(Ei)P(A|Ei) product separately before summing.
5. In random variable problems, always check sum pi = 1, and compute the mean first, then the variance using E(X^2).
6. In binomial problems, first identify n, p, and r from the story: n is the number of trials, p the success probability, and r the required number of successes.
7. Recognise phrases: "at least one" means 1 - P(X = 0), "at most k" means sum of P(X = r) from 0 to k, and "exactly k" means the single term P(X = k).
8. Practise the standard Bayes' problem with two or three boxes/bags and a drawing, since this exact template recurs in the examination.

Conclusion

Probability generalises intuition about chance into a rigorous calculus of uncertainty. Conditional probability refines likelihood in the light of new information, the multiplication rule and independence structure the joint behaviour of events, and the law of total probability and Bayes' theorem reverse the flow of conditioning to update beliefs from evidence. Random variables turn experiments into numerical data, with the mean and variance summarising a distribution, and the binomial distribution models the counting of successes in repeated trials with remarkable accuracy and elegance. These tools extend far beyond the classroom into statistics, science, and decision-making. Because the concepts are subtle but the methods are formulaic once the problem is correctly interpreted, careful labelling of events and systematic application of the standard formulas guarantee strong performance in this highest-weightage chapter.