Probability revision notes

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.

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.

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 $A$, $P(A) = \sum_{\omega_i \in A} P(\omega_i)$.

• 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 $\sigma^2 = \sum x_i^2 p_i - \mu^2$.

(Note: Probability is a dimensionless, unitless ratio. Always falls in the interval $[0, 1]$.)

• 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 $P(B) = 0$.

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: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

• Mutually Exclusive vs. Independent: Students frequently confuse these. Mutually exclusive means $P(A \cap B) = 0$. Independent means $P(A \cap B) = P(A)P(B)$. Non-empty mutually exclusive events are never independent.
• "Or" vs. "And": "Or" correlates with Union ($\cup$) and Addition. "And" correlates with Intersection ($\cap$) and Multiplication.

• Pairwise vs. Mutually Independent: For 3 events to be independent, they must be pairwise independent ($P(A \cap B) = P(A)P(B)$, etc.) AND mutually independent ($P(A \cap B \cap C) = P(A)P(B)P(C)$). Pairwise independence does not guarantee mutual independence.
• Order Importance in Combinatorics: When drawing items "without replacement", standard combination formulas ($^nC_r$) ignore order. If order matters (e.g., drawing specifically a Red then Blue), use permutations ($^nP_r$) or direct multiplication of sequential probabilities.

• 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.