In short: Use the total probability theorem when you want the overall probability of an effect across several causes — it adds up P(effect|cause)·P(cause). Use Bayes' theorem when you already know the effect happened and want the probability of a particular cause. Bayes is built on top of total probability: it uses that same sum as its denominator.
The choice in total probability theorem vs Bayes theorem comes down to the direction of the question. Total probability runs forward — causes to effect — and answers "how likely is this outcome overall?" Bayes runs backward — effect to cause — and answers "given the outcome, which cause was responsible?" JEE Advanced often stacks them in one problem: first you compute a total probability, then you feed it straight into Bayes. Knowing which tool the question is asking for saves you from doing the wrong calculation perfectly.
The reliable method
Decide between the two with this procedure:
- Find the direction of the question. Are you asked for the chance of the effect (forward) or the chance of a cause given the effect (backward)?
- Forward → total probability. If you want P(effect), sum over the causes: P(E) = Σ P(E|Cᵢ)·P(Cᵢ).
- Backward → Bayes. If you are told the effect happened and want P(Cₖ|E), use P(Cₖ|E) = P(E|Cₖ)·P(Cₖ) / P(E).
- Notice they share a denominator. The P(E) inside Bayes is the total probability — so a backward question usually requires the forward calculation first.
- Match the answer to the stem. "What is the probability it is defective?" is total probability; "given it is defective, which machine?" is Bayes.
A worked example
Two bags are on a table. Bag A holds 4 white and 6 black balls; Bag B holds 7 white and 3 black balls. A bag is chosen at random, and one ball is drawn. (a) Find the probability the ball is white. (b) Given the ball is white, find the probability it came from Bag A.
Causes: A and B, each chosen with probability 1/2.
Part (a) — total probability (forward):
P(white|A) = 4/10, P(white|B) = 7/10.
P(white) = P(white|A)·P(A) + P(white|B)·P(B) = (4/10)(1/2) + (7/10)(1/2) = 4/20 + 7/20 = 11/20 = 0.55
Part (b) — Bayes (backward):
P(A|white) = P(white|A)·P(A) / P(white) = (4/20) / (11/20) = 4/11 ≈ 0.364
So overall a white ball appears with probability 11/20, and once you see a white ball the chance it came from Bag A is 4/11. The method holds because part (a) builds the total P(white) by summing both causes, and part (b) reuses that exact 11/20 as its denominator — Bayes literally cannot run without the total-probability result.
Common mistakes to avoid
- Using Bayes when the question only wants the effect's probability. Fix: if there is no "given", you likely need total probability, not Bayes.
- Computing total probability but forgetting the final Bayes divide. Fix: a "given the effect" question is not finished until you divide by P(E).
- Recomputing P(E) twice with different values. Fix: the P(E) in Bayes is the same number you found by total probability — carry it forward.
- Treating the causes as overlapping. Fix: both theorems need a partition — mutually exclusive, exhaustive causes summing to 1.
- Mixing up P(E|Cₖ) and P(Cₖ|E). Fix: the forward conditional is a likelihood; the backward one is the posterior Bayes returns.
Frequently asked questions
When do I use total probability versus Bayes' theorem? Use total probability to find the overall chance of an effect across all causes. Use Bayes when the effect is known and you want the probability of a specific cause.
Is Bayes' theorem just total probability rearranged? Bayes uses the total probability result as its denominator. So total probability is the forward step, and Bayes is the backward step that divides a single joint probability by that total.
Can one JEE problem need both theorems? Yes, very often. A typical question asks for P(effect) first (total probability) and then P(cause|effect) (Bayes), reusing the same denominator.
What do the causes have to satisfy for either theorem? They must form a partition: mutually exclusive and exhaustive, with prior probabilities summing to 1. Both theorems rely on this.
Which theorem gives a posterior probability? Bayes' theorem. It updates a prior P(cause) into a posterior P(cause|effect) after the evidence is observed; total probability does not produce a posterior.
Practise this
Find out if this trips you up — [take the free diagnostic](/diagnostic), then work through the [free Socratic lesson](/lesson/start) to practise picking the right theorem fast.