Probability TreesJEE Main

Concept

A probability tree lays out a sequence of events so that each path's probability is the product of the branch probabilities along it. Summing the paths that include event $B$ gives the law of total probability, and comparing one path to that total gives Bayes' theorem.

Key formula

Derivation

Each branch from a node carries a conditional probability, and probabilities along a path multiply: $P(A\cap B) = P(A)P(B|A)$.

Different paths are mutually exclusive, so probabilities of paths add. Event $B$ is reached via two paths, giving the total probability formula.

Bayes' theorem then reverses the conditioning: the posterior $P(A|B)$ is the share of $P(B)$ that came through $A$.

Scenarios to explore

• Probability Trees — Conditional probability via tree diagrams.

Real-world applications

• Medical testing — true/false positives and disease prevalence.
• Spam filters and machine-learning classifiers.
• Reliability of multi-stage systems.

JEE exam tips

• Check your tree: all leaf probabilities must sum to exactly 1.
• Bayes is just one joint leaf divided by the total probability of the evidence.

Common mistakes

• Adding branch probabilities along a single path instead of multiplying.
• Confusing $P(A|B)$ with $P(B|A)$ (the prosecutor's fallacy).
• Forgetting that branches from a node must sum to 1.

Exam traps to avoid

• A rare event with a good test can still give a low $P(A|B)$ — base rates matter.
• Conditional probabilities are not symmetric: $P(A|B) \neq P(B|A)$ in general.