CAT Probability Questions with Solutions PDF; Preparation Guide & Practice Questions

Given below are some CAT Probability Combinatorics questions for practice:

Question 1

• Difficulty: Easy

• Question: How many 4-digit numbers, each greater than 1000 and each having all four digits distinct, are there with 7 coming before 3?

• Options: [Type In The Answer (TITA)]

Question 2

• Difficulty: Medium

• Question: A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or a large-sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich is

• Options:

1. 880

2. 840

3. 800

4. 600

Question 3

• Difficulty: Medium

• Question: In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers?

• Options: [Type In The Answer (TITA)]

Question 4

• Difficulty: Hard

• Question: The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons, but no child gets an odd number of balloons, is

• Options: [Type In The Answer (TITA)]

Question 5

• Difficulty: Hard

• Question: Let AB, CD, EF, GH, and JK be five diameters of a circle with centre at O. In how many ways can three points be chosen out of A, B, C, D, E, F, G, H, J, K, and O so as to form a triangle?

• Options: [Type In The Answer (TITA) - No options provided in the official paper]

Read More:

• Best Books for CAT Preparation 2026
• CAT Exam Previous Year Question Papers

In the CAT Quantitative Aptitude section, Probability questions appear under the Modern Math module. Over the last few years, there were around 1 to 2 questions on Probability in each slot. CAT probability questions are often combined with concepts of Permutation and Combination (P&C).

If your basic concepts and counting techniques are clear, Probability questions take less time to solve compared to lengthy Geometry or Algebra problems.

CAT Probability Question Distribution (Recent Years)

Before practising questions from the PDF, review these core definitions, rules, and mathematical formulas.

1. Basic Definitions

• Sample Space (S): The set of all possible outcomes of an experiment.
• Favourable Outcomes (n(E)): The number of outcomes that satisfy the given condition.
• Classical Probability: P(E) = n(E) / n(S), where 0 ≤ P(E) ≤ 1.
• Complementary Event: P(E') = 1 - P(E), representing the probability that event E does not happen.

2. Addition Theorem (Union of Events)

Used when you need the probability of event A or event B occurring:

• For any two events: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Mutually Exclusive Events: Two events that cannot happen at the same time (P(A ∩ B) = 0). Here, P(A ∪ B) = P(A) + P(B).

3. Multiplication Theorem and Independent Events

Used when you need the probability of event A and event B occurring together:

• Independent Events: The occurrence of event A does not change the probability of event B. Formula: P(A ∩ B) = P(A) × P(B).
• Dependent Events: The outcome of the first event affects the outcome of the second (such as drawing items without replacement).

4. Conditional Probability

The probability of event A happening, given that event B has already occurred:

P(A | B) = P(A ∩ B) / P(B), provided P(B) > 0.

5. Quick Combinatorial Setups

• Coins: Tossing n coins results in 2ⁿ total outcomes.
• Dice: Rolling n standard six-sided dice results in 6ⁿ total outcomes.
• Cards: A standard deck has 52 cards (4 suits of 13 cards each; 26 red, 26 black; 12 face cards).
• Selection (P&C linkage): Selecting r items out of n items uses ⁿC_r = n! / [r! × (n - r)!].

Common Traps to Avoid:

• "At Least One" Conditions: Instead of calculating multiple positive cases, use the shortcut: P(At least one) = 1 - P(None).
• Replacement vs. Non-Replacement: Check if drawn items are put back. Non-replacement reduces both the numerator and the denominator on subsequent draws.
• Order Sensitivity: Determine whether order matters. If picking two balls one by one, account for the sequence; if picking them together, treat it as a single combination.

Probability questions become easy when a test taker approaches it systematically. Follow this three-step study workflow to build speed and accuracy.

Step 1: Finish Permutations and Combinations First

Probability is simply counting favourable cases and dividing by total cases. If your concepts of arrangements, selections, circular arrangements, and identical object distributions are incomplete, you will struggle to build the sample space. Always complete your P&C prep before starting Probability.

Step 2: Solve Topic-Wise Practice Sets

Move through question types in order of difficulty:

1. Coins, Dice, and Cards: Practice listing sample spaces cleanly and using basic combinations.
2. Balls in Bags / Item Selection: Master non-replacement cases and conditional constraints.
3. Target and Game Problems: Practice independent probability chains (for example, two people shooting at a target until someone hits).
4. Advanced Logic Sets: Work through problems combining conditional statements and Venn diagrams.

Step 3: Track Accuracy in Timed Mocks

During the exam, you should spend no more than 2 to 2.5 minutes on a Probability question. If you cannot identify the sample space within 45 seconds, flag the question and move on. Regular timed sectionals will help you quickly tell straightforward questions apart from lengthy trap problems.

Find topic-wise CAT prep practice questions below:

Check some important FAQs related to CAT Probability: