JEE Main 2025 October: Conditional probability on a tree
The event asked for was "reached the third box", not "passed all three boxes", and that difference — one word of the question — was the whole exam.
Worked solution for the JEE Main October 2025 probability question, where three sequential boxes had to be read as a probability tree and the final part needed Bayes theorem.
How this actually works
Reading the tree correctly
The question described three sequential boxes, each reached only if the candidate had passed the previous one. That structure is a Markov chain in disguise, and the clean way to solve it is a probability tree with the pass rates written on each branch. The first trap is to multiply the three pass rates straight through: that gives P(all three), which is not the event the question asked for.
The event the question actually asks for
The required probability was that the candidate reached the third box, not that they passed all three. Reaching the third box includes failing box two, because the chain continues only if box one was passed. So the calculation has two branches: pass box one and pass box two, or pass box one and fail box two but still move on. Those two products are added, and the second term is the one that gets dropped.
The Bayes check
The final sentence asked for the probability that, given the candidate did reach box three, they had actually passed box two — a straight Bayes application. The two pieces needed are the numerator, the joint probability of the two passed events, and the denominator, the total probability of reaching box three from the tree above. Most candidates lost marks here by reusing the first answer as the numerator without conditioning it.
The drawing that does the work
The whole question is one diagram. Draw the tree with three stages, write each branch probability at the branch, and mark the paths that satisfy the event with a tick. Every probability question in JEE that looks like algebra is actually a picture, and the candidates who drew the tree got the answer in three minutes while the ones who did not spent twenty.
JEE Main 2025 October: Conditional probability on a tree: the facts to keep
| Question type | Probability, Section A (4 marks) |
|---|---|
| The structure | Three sequential boxes, each reached only if the previous one was passed |
| The event asked for | Reaching the third box, which is not the same as passing all three boxes |
| The two-branch sum | Pass box one and pass box two, or pass box one and fail box two but still advance |
| The Bayes part | The final sentence asks for a conditional probability, so the numerator and denominator must both be conditioned |
| The method | Draw the probability tree with each branch probability written at the branch, then tick the paths that satisfy the event |
Questions students ask about this
How do I know whether to find P(A and B) or P(A given B)?
If the question says "given that", it is conditional and you need Bayes. If it says "what is the chance of both", it is the product and no conditioning is needed. Reading for the word "given" is faster than trying to infer it from the numbers.
Why did the question use three boxes rather than two?
To force the distinction between reaching and passing. With two boxes the two readings give the same answer; with three they diverge, so the question can mark the difference.