🔢 Quant

Probability

Classical probability, addition & multiplication rules.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

1/5

of recent CAT years

% of Quant📈

~0%

of section questions

Est. hours⏱️

~6h

to master

2021

2022

2023

~1/22

2024

2025

🎯PYQ Evidence

CAT 2021–2025: ~0.1 per slot (2024: 0.3). Exactly one pure probability question in five years (2024) — prep it as applied counting, not as a separate ocean.

Probability

Probability is just favourable ÷ total once you count carefully — so it leans heavily on the P&C skills next door. The fast routes are the complement and the tree.

Core ideas

$P(E)=\frac{\text{favourable outcomes}}{\text{total outcomes}},\qquad 0\le P(E)\le 1.$

Complement: $P(\text{not }E)=1-P(E)$ — the engine behind "at least one."
Independent events (AND): $P(A\cap B)=P(A)\,P(B)$ .
Mutually exclusive (OR): $P(A\cup B)=P(A)+P(B)$ .
Conditional: $P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$ .

A worked example

Two fair dice are rolled. What is the probability that the sum is 7?

Total outcomes $=6\times6=36$ . The favourable pairs are $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)$ — six of them:

$P(\text{sum}=7)=\frac{6}{36}=\frac16.$

Seven is the most likely sum precisely because it has the most pairs — a fact worth remembering for dice questions.

🎯PYQ Evidence

When everyone does at least one of two things, inclusion-exclusion turns "both" into a single linear function whose extremes sit at the endpoints. : since every student does at least one activity, by inclusion-exclusion the share doing both = swimming-share + running-share − 100% within each group; summing boys and girls makes the total "both" a straight-line function of the girl count G. Because the count is linear in G, its minimum and maximum land at the boundaries of the allowed 44%–60% range — giving 72 and 80. The takeaway: write the overlap as both = A + B − total, then push the free variable to its limits.

Common traps

Adding non-exclusive probabilities. If $A$ and $B$ can both happen, subtract the overlap: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ .
Independent vs mutually exclusive — opposite ideas: exclusive events can't co-occur, independent ones don't influence each other.
With vs without replacement changes the denominator on the second draw.

Checklist

Count total and favourable with care (P&C)
Use $1-P(\text{none})$ for "at least one"
Multiply for independent, add for mutually exclusive
Adjust for replacement between draws

Sample Questions

32 practice questions

Easy

In a group of 500 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 70% of the girls opted for swimming while 60% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

Easy

In a group of 250 students, the percentage of girls was at least 24% and at most 56%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 60% of the girls opted for swimming while 80% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2024 · Slot 3

Hard

In a group of 250 students, the percentage of girls was at least 44% and at most 60%. The rest of the students were boys. Each student opted for either swimming or running or both. If 50% of the boys and 80% of the girls opted for swimming while 70% of the boys and 60% of the girls opted for running, then the minimum and maximum possible number of students who opted for both swimming and running, are

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