CAT Quant · Study & Practice

Probability

AreaModern Maths DifficultyModerate CAT weightage1–2 questions (often paired with Permutations & Combinations and Set theory)

Probability is the measure of how likely an event is, expressed as a number between 0 (impossible) and 1 (certain). For CAT, XAT and SNAP it sits at the heart of Modern Maths, and the questions are rarely about memorising a formula — they reward students who can count cleanly. Almost every probability problem reduces to two counts: the number of favourable outcomes and the number of total equally likely outcomes, and the ratio of the two is the answer. That is why probability leans so heavily on Permutations & Combinations: get the counting right and the probability falls out in one line. This chapter builds the chain step by step — classical probability with dice, cards and balls; the addition and complement rules for "or" and "at least" questions; conditional probability for "given that" situations; independence for events that do not influence each other; and a gentle first look at Bayes’ theorem, which reverses a conditional to update a belief after new evidence. Throughout, the focus is on the fast, exam-safe method: define the sample space, count favourable cases with C(n,r), and use the complement whenever "at least one" appears.

Topics

1 Classical Probability 4 worked · 5 practice · 8-Q test Start → 2 Conditional Probability 4 worked · 5 practice · 8-Q test Start → 3 Independent Events 4 worked · 5 practice · 8-Q test Start → 4 Bayes Basics 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • For any "at least one" question use the complement: P(at least one) = 1 − P(none). It almost always counts faster.
  • Fix the sample space before counting: two dice = 36, a pack = 52, "with replacement" keeps the denominator constant.
  • When items are drawn together, count with combinations C(n,r); when drawn in order or with replacement, multiply probabilities.
  • Addition rule for "A or B": P(A) + P(B) − P(A∩B). Drop the overlap term only when the events are mutually exclusive.
  • Independent ⇒ multiply, P(A∩B) = P(A)·P(B). Mutually exclusive ⇒ add, P(A∩B) = 0. Never mix the two up.
  • For Bayes, draw a two-branch tree: multiply along the favourable branch, then divide by the sum of all branches.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Confusing independent events (can co-occur, multiply) with mutually exclusive events (cannot co-occur, add).
  • Forgetting to subtract the overlap P(A∩B) in the addition rule, so "A or B" is overcounted.
  • Treating "without replacement" draws as independent — the second probability changes after the first removal.
  • Inverting the conditional: using P(A|B) when the question asks for P(B|A) — this is exactly what Bayes corrects.
  • Miscounting the sample space, e.g. taking two dice as 12 or 30 outcomes instead of 36.

📈 CAT exam insight & PYQ analysis

In CAT, probability typically contributes 1–2 questions and is heavily intertwined with Permutations & Combinations and Set theory, so clean counting decides the marks more than any formula. The recurring patterns are dice and card setups solved by the complement ("at least one"), without-replacement sequential draws using the multiplication rule, and conditional/"given that" problems. XAT and SNAP push a little harder, occasionally testing a Bayes-style two-source problem (two bags or two machines) or expected value. The smart preparation is to master combination counting and the complement shortcut first; once the count is reliable, the probability is a one-line ratio.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Classical probability formula?Tap to reveal
P(E) = favourable / total
Range of any probability?Tap to reveal
0 ≤ P(E) ≤ 1
Complement rule?Tap to reveal
P(E′) = 1 − P(E)
Addition rule for P(A∪B)?Tap to reveal
P(A) + P(B) − P(A∩B)
Condition for mutually exclusive?Tap to reveal
P(A∩B) = 0
Conditional probability P(A|B)?Tap to reveal
P(A∩B) / P(B)
Condition for independent events?Tap to reveal
P(A∩B) = P(A)·P(B)
Outcomes when two dice are rolled?Tap to reveal
36
P(sum = 7) with two dice?Tap to reveal
6/36 = 1/6
P(at least one six in two rolls)?Tap to reveal
1 − (5/6)² = 11/36
Number of face cards in a pack?Tap to reveal
12 (J, Q, K × 4 suits)
Bayes’ theorem for P(B|A)?Tap to reveal
P(B)·P(A|B) / P(A)

📌 Quick revision

Probability is favourable over total, always between 0 and 1. Use the complement for "at least one" (1 − P(none)) and the addition rule P(A∪B) = P(A) + P(B) − P(A∩B) for "or", subtracting the overlap. Count together-draws with combinations and order/with-replacement draws by multiplying. Conditional probability P(A|B) = P(A∩B)/P(B) shrinks the sample space to B; independent events satisfy P(A∩B) = P(A)·P(B). Bayes’ theorem reverses a conditional, P(B|A) = P(B)·P(A|B)/P(A), best done with a tree: favourable branch over the sum of all branches. Get the counting clean and the answer follows.

Chapter test

📝 Probability — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered Probability when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards