Probability Questions for the CAT exam are part of the Modern Mathematics section in the CAT exam. Through Algebra questions, aspirants are tested on concepts related to Formula, Concepts, Permutations. The difficulty level of the Algebra questions can be easy to moderate.

## Tips to improve Probability Questions for CAT?

- Solve a wide range of probability problems, including conditional probability, Bayes' theorem, expected value, variance, and distributions.
- Strengthen Calculation Skills
- Solve Mock Tests and Previous Years' Papers
- Review and Learn from Mistakes

## Probability Questions Tricks for the CAT Exam

- Pay close attention to the details provided in the problem statement. Identify the events, conditions, and probabilities mentioned.
- For problems involving arrangements or selections, use combinations and permutations to calculate probabilities efficiently.
- Start with questions that appear easier or require less calculation time to secure marks early.

## Probability Questions Concepts for the CAT Exam

Sample Space (S):

The set of all possible outcomes of an experiment.

Example: For a coin toss, S={H, T}.

Event (E):

A subset of the sample space.

Example: Getting a head when tossing a coin.

Probability of an Event (P(E)):

The measure of the likelihood that an event will occur.

Formula: P(E) = (Number of favorable outcomes) / (Total number of outcomes)

**Probability Practice Questions for the CAT exam**

**Q1**. The sum of the perimeters of an equilateral triangle and a rectangle is 90cm. The area, T, of the triangle and the area, R, of the rectangle, both in sq cm, satisfy the relationship. If the sides of the rectangle are in the ratio 1:3, then the length, in cm, of the longer side of the rectangle, is

A. 27

B. 21

C. 24

D. 18

**Level: Moderate**

**Answer: **C

**Q2**. Let C be a circle of radius 5 meters having center at O. Let PQ be a chord of C that passes through points A and B where A is located 4 meters north of O and B is located 3 meters east of O. Then, the length of PQ, in meters, is nearest to

A. 8.8

B. 7.8

C. 6.6

D. 7.2

**Level: Difficult**

**Answer: **A

**Q3.** The vertices of a triangle are (0,0), (4,0) and (3,9). The area of the circle passing through these three points is

**Level: Easy**

**Answer:** 85π/9

**Q4.** The points (2,1) and (-3,-4) are opposite vertices of a parallelogram. If the other two vertices lie on the line x + 9y + c = 0, then c is

A. 12

B. 13

C. 15

D. 14

**Level: Easy**

**Answer:** Option D

**Q5.** A circle is inscribed in a rhombus with diagonals 12 cm and 16 cm. The ratio of the area of circle to the area of rhombus is

**Level: Difficult**

**Answer:** 6π/25

**What are the must-do Probability questions for the CAT exam?**

**Q6**. On a rectangular metal sheet of area 135 sq in, a circle is painted such that the circle touches two opposite sides. If the area of the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is

**Level: Moderate**

**Answer:** 6π/25

**Q7.** A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

A. 243

B. 232

C. 256

D. 264

**Level: Easy**

**Answer:** 6π/25

**Q8.** If the volume of a right circular cylinder that can be cut out from a sphere of radius 6 cm is maximum possible, then find the radius of the cylinder?

**Level: Easy**

**Answer:** 6π/25

**Q9.** A solid sphere is cut into 8 identical pieces by three mutually perpendicular cuts. By what percentage is the sum of the total surface areas of the eight pieces more than the total surface area of the original sphere?

A. 110

B. 150

C. 70

D. 90

**Level: Difficult**

**Answer:** 6π/25

**Q10.** What is the number of distinct points of intersection of x^2 + y^2 = 169, 5x + 12y = 169 and x = 0?

A. 2

B. 4

C. 3

D. 1

**Level: Moderate**

**Answer:** Option B

**What are the most difficult Probability questions for the CAT 2024 exam?**

**Q11. **The graph of y = x^2 − 8x + 13 is symmetric wrt to the line x = c, then the value of c is

**Level: Easy**

**Answer:** 6π/25

**Q12.** In an equilateral triangle ABC, a point P is marked on AB. The side AC is extended to Q such that APQ is a right triangle with P = 90. The area of ABC is equal to the area of APQ. If BP= then find the value of AB?

**Level: Difficult**

**Q13. **All the three sides of an equilateral triangle ABC are divided into 3 equal parts using 2 points on each side. Each side has length 6 cm. The points on side AB are A1 and A2 with point A1 being the nearest to A. The points on side BC are B1 and B2 with point B1 being the nearest to B. The points on side AC are C1 and C2 with point C1 being the nearest to C. A circle is drawn inside A1B1C1 such that it touches all its sides. Find the radius of the circle.

**Level: Moderate**

**Answer:** 6π/25

**Q14.** A trapezium ABCD has sides AD and BC parallel to each other. The diagonal AC divides the trapezium into two parts such that ABC is an equilateral triangle and ACD is a right triangle right angled at C. If the area of ABC is 100 square cm, then the area of the trapezium (in square cm) is

**Level: Easy**

**Answer:** 6π/25

**Q15.** In a right triangle ABC, ∠ ABC is 90 âˆ˜. The ratio of sides AB and AC is 2:5. What is the ratio of side BC and the circumradius of the triangle?

**Level: Difficult**

**Answer:** 84:5

**Find some previous year Probability questions from the CAT 2023**

**Q16.** PQRXYZ is a hexagon in which all its interior angles are equal. If the lengths of PQ= 1 cm, QR = 4 cm, RX = 2 cm, XY = 2 cm. Then lengths of PZ and ZY are

A. 3, 3

B. 4, 2

C. 2, 4

D. 4, 4

**Level: Easy**

**Answer:** 6π/25

**Q17.** What is the number of distinct triangles with integral valued sides and perimeter 14?

A. 6

B. 5

C. 4

D. 3

**Level: Difficult**

**Q18.** ABCD is a rhombus with the diagonals AC and BD intersecting at the origin on the x-y plane. The equation of the straight line AD is x + y = 1. What is the equation of BC?

A. x + y = –1

B. x – y = –1

C. x + y = 1

D. None of these

**Level: Moderate**

**Q19.** A farmer has decided to build a wire fence along one straight side of his property. For this, he planned to place several fence-posts at 6 m intervals, with posts fixed at both ends of the side. After he bought the posts and wire, he found that the number of posts he had bought was 5 less than required. However, he discovered that the number of posts he had bought would be just sufficient if he spaced them 8 m apart. What is the length of the side of his property and how many posts did he buy?

A. 100 m, 15

B. 100 m, 16

C. 120 m, 15

D. 120 m, 16

**Level: Easy**

**Answer:** 6π/25

**Q20.** A ladder leans against a vertical wall. The top of the ladder is 8 m above the ground. When the bottom of the ladder is moved 2 m farther away from the wall, the top of the ladder rests against the foot of the wall. What is the length of the ladder?

A. 10 m

B. 15 m

C. 20 m

D. 17 m

**Level: Difficult**

**Answer:** Option D

**What are the most important Probability questions for the CAT 2024?**

**Q 21. **Four horses are tethered at four corners of a square plot of side 14 m so that the adjacent horses can just reach one another. There is a small circular pond of area 20 m² at the centre. Find the ungrazed area.

A. 22 m²

B. 42 m²

C. 84 m²

D. 168 m²

**Level: Moderate**

**Q 22. **Neeraj has agreed to mow a lawn, which is a 20 m × 40 m rectangle. He mows it with 1 m wide strip. If Neeraj starts at one corner and mows around the lawn toward the centre, about how many times would he go round before he has mowed half the lawn?

A. 2.5

B. 3.5

C. 3.8

D. 4

**Level: Easy**

**Q 23. **Find the minimum integral value of n such that the division 5n/124 leaves no remainder.

A. 124

B. 123

C. 31

D. 62

**Level: Difficult**

**Q 24. **It is desired to extract the maximum power of 3 from 24! where n! = (n-1)(n-2)...3 2 1. What will be the exponent of 3?

**Level: Moderate**

**Q 25. **If a < b, which of the following is always true?

A. a < (a + b) / 2 < b

B. a < ab / 2 < b

C. a < b² - a² < b

D. a

**Level: Difficult**

**Answer:** Option C

**Which are the most repeating Probability questions in the CAT Exam?**

**Q 26.** What is the highest power of 3 available in the expression 58! - 38!

A. 17

B. 18

C. 19

D. None of these

**Level: Moderate**

**Q 27.** If the real root of the cubic equation 8a³ - 12a² - 6a - 1 = 0 is expressed as (p^(1/3) + q^(1/3) + 1) / r where p, q, r are natural numbers, what is the value of p + q + r?

**Level: Easy**

**Q 28.** In how many ways can a pair of integers (x, a) be chosen such that x² - 2|x| + |a - 2| = 0?

A. 6

B. 5

C. 4

D. 7

**Level: Difficult**

**Answer:** Option D

**Q 29.** The number of pairs of integers (x, y) satisfying x ≥ y ≥ -20 and 2x + 5y = 99

**Level: Easy**

**Answer:** 17

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