Probability revision notes

Key Concepts & Definitions

Probability theory attempts to quantify the chances of occurrence or non-occurrence of events. The modern axiomatic approach was introduced by A. N. Kolmogorov in 1933.

Sample Space (SSS)

The universal set containing all possible outcomes of a random experiment.

Event (EEE)

Any subset EEE of a sample space SSS is called an event. The event EEE is said to have occurred if the outcome ω\omegaω of the experiment is such that ω∈E\omega \in Eω∈E.

Impossible Event

The empty set ϕ\phiϕ, which contains no sample points, is an event that cannot occur.

Sure Event

The entire sample space SSS is considered a sure event because the outcome will always belong to SSS.

Simple (Elementary) Event

An event containing exactly one sample point of the sample space. In a sample space of nnn elements, there are exactly nnn simple events. → [JEE TIP] Simple events of a sample space are always mutually exclusive.

Compound Event

An event containing more than one sample point.

Mutually Exclusive Events

Two events AAA and BBB are mutually exclusive if they cannot occur simultaneously, meaning they are disjoint sets (A∩B=ϕA \cap B = \phiA∩B=ϕ).

Exhaustive Events

A set of events E1,E2,...,EnE_1, E_2, ..., E_nE1​,E2​,...,En​ is exhaustive if their union is the entire sample space (E1∪E2∪...∪En=SE_1 \cup E_2 \cup ... \cup E_n = SE1​∪E2​∪...∪En​=S). At least one of them necessarily occurs whenever the experiment is performed.

Mutually Exclusive and Exhaustive Events

A set of events that are pairwise disjoint (Ei∩Ej=ϕE_i \cap E_j = \phiEi​∩Ej​=ϕ for i≠ji \neq ji=j) and whose union forms the sample space SSS.

Algebra of Events and Set Theory Equivalences

Analogous to set theory, events can be combined to form new events:

• Complementary Event ('not A'): Denoted by $A'$, $\overline{A}$, or $A^c$. It consists of all outcomes in $S$ that are not in $A$. $A' = \{\omega: \omega \in S \text{ and } \omega \notin A\} = S - A$.
• The Event 'A or B' ($A \cup B$): The event that either $A$ occurs, or $B$ occurs, or both occur.
• The Event 'A and B' ($A \cap B$): The event that both $A$ and $B$ occur simultaneously.
• The Event 'A but not B' ($A - B$): Elements in $A$ but not in $B$. Represented as $A - B = A \cap B'$. → [JEE TIP] The formula $P(A - B) = P(A) - P(A \cap B)$ is heavily tested in JEE for calculating exact occurrences.

Axiomatic Approach to Probability

Let $S$ be the sample space. The probability $P$ is a real-valued function whose domain is the power set of $S$ and range is the interval $[0, 1]$, satisfying the following axioms:

1. Axiom 1: For any event $E$, $P(E) \ge 0$.
2. Axiom 2: $P(S) = 1$.
3. Axiom 3: If $E$ and $F$ are mutually exclusive events, then $P(E \cup F) = P(E) + P(F)$.

From Axiom 3, substituting $F = \phi$, we get $P(E \cup \phi) = P(E) + P(\phi) \Rightarrow P(\phi) = 0$. For outcomes $\omega_i \in S$:

• $0 \le P(\omega_i) \le 1$.
• $\sum P(\omega_i) = 1$.
• For any event $A$, $P(A) = \sum_{\omega_i \in A} P(\omega_i)$.

Advanced Probability Concepts (JEE Advanced Topics)

• Conditional Probability: The probability of event $A$ occurring given that event $B$ has already occurred. $P(A|B) = \frac{P(A \cap B)}{P(B)}$ (where $P(B) \neq 0$).
• Multiplication Theorem: $P(A \cap B) = P(A) \cdot P(B|A)$.
• Independent Events: Two events $A$ and $B$ are independent if the occurrence of one does not affect the occurrence of the other. Condition: $P(A \cap B) = P(A) \cdot P(B)$. → [JEE TIP] Do not confuse independent events with mutually exclusive events. Mutually exclusive events are highly dependent (if one happens, the other strictly cannot).
• Law of Total Probability: If $E_1, E_2, ..., E_n$ form a set of mutually exclusive and exhaustive events, and $A$ is any event, then $P(A) = \sum_{i=1}^{n} P(E_i) \cdot P(A|E_i)$.
• Bayes' Theorem: Reverses conditional probability. $P(E_i|A) = \frac{P(E_i) \cdot P(A|E_i)}{\sum_{j=1}^{n} P(E_j) \cdot P(A|E_j)}$. → [JEE TIP] Always define your $E_i$ partitions clearly before applying Bayes' Theorem to avoid denominator traps.
• Binomial Probability Distribution: For $n$ independent Bernoulli trials with probability of success $p$ and failure $q = 1-p$, the probability of exactly $r$ successes is $P(X=r) = {}^nC_r p^r q^{n-r}$.
• Random Variables: Mean (Expected value) $\mu = \sum x_i p_i$. Variance $\sigma^2 = \sum x_i^2 p_i - \mu^2$.

Conditions & Limitations

• Equally Likely Restriction: The classical formula $P(E) = n(E) / n(S)$ CANNOT be used if the sample points are not equally likely. If a coin is biased, you must use the Axiomatic rule $P(A) = \sum_{\omega_i \in A} P(\omega_i)$ by summing the individual valid, non-equal probabilities.
• Axiomatic Validity: Any assigned probability distribution is only valid if both $0 \le P(\omega_i) \le 1$ and $\sum P(\omega_i) = 1$ hold true simultaneously.
• Empty Sets in Conditionals: $P(A|B)$ is completely undefined if $P(B) = 0$.

Standard Derivations & Step-by-Step Problem Solving

Derivation of the Addition Theorem $P(A \cup B) = P(A) + P(B) - P(A \cap B)$:

1. Express $A \cup B$ as disjoint sets: $A \cup B = A \cup (B - A)$.
2. Because $A$ and $(B - A)$ are mutually exclusive, apply Axiom 3: $P(A \cup B) = P(A) + P(B - A)$.
3. Express $B$ as disjoint sets: $B = (A \cap B) \cup (B - A)$.
4. Apply Axiom 3 again: $P(B) = P(A \cap B) + P(B - A)$.
5. Subtract the second equation from the first: $P(A \cup B) - P(B) = P(A) - P(A \cap B)$.
6. Rearrange to get the final theorem: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Previous Year JEE Topics

• Bayes' Theorem & Total Probability: Heavily tested via urn models, diagnostic tests, and sequential factory production scenarios.
• Binomial Distribution bounds & maximization: Finding the maximum probability in a binomial distribution or calculating expected values.
• Geometric Probability: Solving probability of regions intersecting (using area/volume integrals instead of discrete counting).
• Derangements: Probability that none of $n$ items goes into their correct corresponding envelopes.
• Sets and Venn Diagrams: Extracting $P(A \cap B')$, $P(A \cup B)'$, etc., given percentages of a population.

Top 10 JEE MCQ Traps

1. Misconception: Assuming $P(A \cup B) = P(A) + P(B)$ universally. Correct Understanding: This only applies if $A$ and $B$ are strictly mutually exclusive ($A \cap B = \phi$). Always subtract $P(A \cap B)$ otherwise.
2. Misconception: Treating Mutually Exclusive and Independent events as the same thing. Correct Understanding: Mutually exclusive means $P(A \cap B) = 0$. Independent means $P(A \cap B) = P(A)P(B)$.
3. Misconception: Calculating "probability of A or B but not both" as $P(A \cup B)$. Correct Understanding: "Exactly one of A or B" requires subtracting the intersection twice: $P(A \cup B) - P(A \cap B) = P(A) + P(B) - 2P(A \cap B)$.
4. Misconception: Multiplying probabilities directly without checking for replacement. Correct Understanding: If drawing without replacement, the sample space shrinks. You must use conditional probability $P(B|A)$ or Combinatorics ($^nC_r$).
5. Misconception: Summing $P(E_i)$ in a Bayes' theorem denominator without making sure the events $E_i$ form an exhaustive and mutually exclusive partition. Correct Understanding: The Law of Total Probability only holds if $\sum P(E_i) = 1$ and $E_i \cap E_j = \phi$.
6. Misconception: Ignoring the scaling factor when a problem states a restricted sample space (e.g., "Given that an even number rolled..."). Correct Understanding: This restricts the denominator. $P(A|B) = n(A \cap B) / n(B)$, not $n(A \cap B) / n(S)$.
7. Misconception: Forgetting that identical objects distribute differently than distinct objects. Correct Understanding: Ensure the classical definition $P = n(E)/n(S)$ uses a sample space of equally likely outcomes. Combinatorics with identical items often creates outcomes that are not equally likely unless items are treated as artificially distinct.
8. Misconception: Assuming $P(A') = 1 + P(A)$ or sign errors in De Morgan's equations. Correct Understanding: $P(A') = 1 - P(A)$. Additionally, $P(\text{neither A nor B}) = 1 - P(A \cup B)$, not $1 - P(A \cap B)$.
9. Misconception: Thinking probability values can be greater than 1 when adding multiple sets. Correct Understanding: Probabilities strictly live in $[0, 1]$. If an addition yields $> 1$, you forgot to subtract the intersection overlaps.
10. Misconception: Confusing $P(A|B)$ with $P(B|A)$ in conditional word problems (Base Rate Fallacy). Correct Understanding: $P(A|B) \neq P(B|A)$. "Probability of having the disease given you tested positive" is distinct from "Probability of testing positive given you have the disease." Use Bayes' Theorem to convert them.