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Permutations & Combinations
Every CAT Permutations & Combinations formula on one page — circular arrangements, restrictions, derangements, and stars and bars — each with a worked example.
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Probability
CAT'26 QUANT CHEATSHEET
Every probability formula and CAT shortcut you need for CAT 2026 — on one page.
Probability on CAT is arithmetic wearing a story — cards, dice, coins and boxes are just vehicles for the same handful of rules: add for OR, multiply for AND, and use the complement whenever “at least one” shows up. Once you can tell independent from mutually exclusive and know when a problem is secretly conditional, most questions become a direct substitution. This sheet lays out every formula you need for the CAT quant section: the addition and multiplication rules, conditional probability and Bayes’ theorem, counting-based probability, binomial probability and expected value, and the union of three events, each with a worked example in real numbers. Keep it open while you practise, and after a mock check where you stand on the CAT score predictor to see which idea is costing you marks.
Probability: every formula you need
1Basic Probability
The fraction of outcomes that satisfy the event of interest.
P(E) = favourable outcomes ÷ total outcomes, 0 ≤ P(E) ≤ 1
Example: rolling a die, P(even) = 3/6 = 1/2.
CAT Hack: Whenever a question asks for “at least one,” compute the complement instead — it's almost always faster.
2Addition Rule (Mutually Exclusive)
Events that cannot happen at the same time just add.
P(A ∪ B) = P(A) + P(B), if A and B cannot both occur
Example: die roll, P(1 or 6) = 1/6+1/6 = 1/3.
3Addition Rule (General)
For overlapping events, subtract the double-counted intersection.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Example: one card, P(King or Heart) = 4/52+13/52−1/52 = 16/52 = 4/13.
Common Mistake: Forgetting to subtract the overlap double-counts the outcomes that satisfy both events.
4Multiplication Rule (Independent Events)
When one event's outcome has no effect on the other's probability.
P(A ∩ B) = P(A) × P(B), if A and B are independent
Example: two coin tosses, P(both heads) = 1/2·1/2 = 1/4.
CAT Insight: Independent and mutually exclusive are opposite ideas — two mutually exclusive events are actually dependent, since one occurring rules the other out.
5Conditional Probability
The probability of A, given that B has already happened.
P(A|B) = P(A ∩ B) ÷ P(B)
Example: 2 cards drawn without replacement, P(2nd is King | 1st is King) = 3/51.
CAT Hack: Any “without replacement” problem is really a conditional probability in disguise — the sample space shrinks after the first draw.
6Complementary Events
The probability of an event not happening.
P(A′) = 1 − P(A)
Example: P(at least 1 head in 3 tosses) = 1−(1/2)3 = 1−1/8 = 7/8.
CAT Favourite: “At least one” over repeated independent trials is one of the most reused CAT setups — always reach for the complement.
7Bayes' Theorem
Reverses a conditional probability using the total probability of B.
P(A|B) = P(B|A)·P(A) ÷ P(B)
Example: P(Red)=0.6 from two boxes → P(Box1|Red) = (0.4·0.5)/0.6 = 1/3.
CAT Insight: Draw a probability tree first — reading Bayes' theorem off the branches is far less error-prone than plugging into the raw formula.
8Probability via Counting
For equally likely outcomes, probability reduces to a ratio of combinations.
P(E) = ⁿCr (favourable) ÷ ⁿCr (total)
Example: 2 balls drawn from 5 red + 3 blue, P(both red) = C(5,2)/C(8,2) = 10/28 = 5/14.
CAT Hack: Card- and ball-drawing problems are usually just an nCr ratio — no probability tree needed.
9Mutually Exclusive vs Independent
Two genuinely different relationships that are easy to mix up.
mutually exclusive: P(A ∩ B)=0 | independent: P(A ∩ B)=P(A)P(B)
Example: “rolling a 1” and “rolling a 6” on one die are mutually exclusive, not independent.
Common Mistake: Mutually exclusive events are never independent (unless one has probability 0) — don't assume the two properties go together.
10At Least One Event
The complement of “none of the trials succeed.”
P(at least 1 success in n trials) = 1 − (1−p)n
Example: P(at least one 6 in 4 rolls) = 1−(5/6)4 = 671/1296 ≈ 0.518.
11Binomial Probability
The probability of exactly k successes in n independent trials.
P(X=k) = ⁿCk pk (1−p)n−k
Example: coin tossed 5 times, P(exactly 3 heads) = C(5,3)(1/2)³(1/2)² = 10/32 = 5/16.
12Expected Value
The long-run average outcome, weighted by probability.
E(X) = Σ x · P(x)
Example: fair die, E(X) = (1+2+3+4+5+6)/6 = 3.5.
CAT Insight: The expected value doesn't need to be an achievable outcome itself — 3.5 is never actually rolled on a die.
13Odds in Favour & Against
A ratio form of probability, common in word problems.
odds in favour = P(E) : P(E′) | odds against = P(E′) : P(E)
Example: P(E)=2/5 → odds in favour = 2:3, odds against = 3:2.
14Probability with Dice & Coins
The classic combined-event setups that recur across CAT questions.
two dice: count ordered pairs (i, j), not unordered sums, out of 36 total
Example: two dice, P(sum=7) = 6/36 = 1/6 (6 ordered pairs sum to 7).
CAT Hack: For two-dice sums, always count ordered pairs like (1,6) and (6,1) separately — there are 36 total outcomes, not 21.
15Union of Three Events
Inclusion-exclusion extended from two events to three.
P(A∪B∪C) = P(A)+P(B)+P(C)−P(A∩B)−P(B∩C)−P(A∩C)+P(A∩B∩C)
Example: 100 people; 50, 40, 30 read three papers, overlaps 20,15,10, all three 5 → at least one = 80.
Common Mistake: Forgetting the final +P(A∩B∩C) term after subtracting all three pairwise intersections under-counts the people in all three groups.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- P(E) = favourable ÷ total; use the complement for “at least one”
- Mutually exclusive → add; overlapping events → add then subtract the intersection
- Independent events multiply; conditional probability divides by the given event
- At least one success in n independent trials = 1 − (1−p)ⁿ
- Binomial: P(X=k) = ⁿCkpk(1−p)n−k
- Card/ball draws: use an nCr ratio directly instead of a full probability tree
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Most Common CAT Traps
- Forgetting to subtract the overlap in the general addition rule (double-counting).
- Assuming mutually exclusive events are also independent — they're opposite relationships.
- Treating a “without replacement” draw as independent instead of conditional.
- Dropping the final +P(A∩B∩C) term when finding the union of three events.
- Expecting the expected value to be an outcome that can actually occur.
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30-Second Revision Box
- P(E) = favourable ÷ total; P(A′) = 1−P(A)
- P(A∪B)=P(A)+P(B)−P(A∩B); independent → P(A∩B)=P(A)P(B)
- P(A|B) = P(A∩B) ÷ P(B)
- At least one = 1 − (1−p)ⁿ
- Binomial P(X=k)=nCk·pᵀ·(1−p)ⁿ⁻ᵀ; E(X)=Σx·P(x)
- Bayes: P(A|B) = P(B|A)P(A) ÷ P(B)
This topic rewards identifying the right rule over grinding through casework — once you see whether events are independent, mutually exclusive, or conditional, the calculation is usually one line. Drill this sheet until the complement trick and the addition rule are reflex, then test them on full sets and track progress with the CAT score predictor. It pairs directly with the Permutations & Combinations cheat sheet, since counting-based probability leans on the same nCr formulas. For more guides, browse the Optima Learn blog or explore every study guide, work through the CAT exam hub, and when you want mentor-led prep, book a free CAT 2026 call.
