Probability Practice Test Questions – Aptitude -- Free Practice

Question 1

A single die is rolled once. Find the probability of getting a number greater than 4.

A

1/3✓✗

B

1/6✓✗

C

1/2✓✗

D

2/3✓✗

Explanation

Total outcomes = 6 (numbers 1 to 6).
Numbers greater than 4: 5, 6 → 2 favourable outcomes.
Probability = 26 = 13.

Question 2

Two coins are tossed together. Find the probability of getting exactly one head.

A

1/4✓✗

B

1✓✗

C

3/4✓✗

D

1/2✓✗

Explanation

Total outcomes when 2 coins are tossed = 4 (HH, HT, TH, TT).
Exactly one head: HT, TH → 2 favourable outcomes.
Probability = 24 = 12.

Question 3

A card is drawn at random from a well-shuffled pack of 52 playing cards. Find the probability that it is a face card (king, queen or jack).

A

1/13✓✗

B

3/13✓✗

C

4/13✓✗

D

1/4✓✗

Explanation

Total cards = 52.
Face cards (king, queen, jack) = 3 per suit × 4 suits = 12.
Probability = 1252 = 313.

Question 4

A bag contains 4 red balls and 6 blue balls. If one ball is drawn at random, find the probability that it is red.

A

2/5✓✗

B

3/5✓✗

C

1/5✓✗

D

1/2✓✗

Explanation

Total balls = 4 + 6 = 10.
Red balls = 4.
Probability = 410 = 25.

Question 5

The probability that a student passes a test is 0.65. Find the probability that the student fails the test.

A

0.65✓✗

B

0.45✓✗

C

0.55✓✗

D

0.35✓✗

Explanation

By the complement rule, P(fail) = 1 - P(pass).
P(fail) = 1 - 0.65 = 0.35.

Question 6

When a fair die is tossed once, find the probability of getting an even number or a number exceeding 4.

A

1/3✓✗

B

1/2✓✗

C

2/3✓✗

D

5/6✓✗

Explanation

Even numbers on a die: 2, 4, 6 (3 outcomes).
Numbers greater than 4: 5, 6 (2 outcomes).
Common to both lists: 6 (1 outcome), so it must be subtracted once to avoid double-counting.
Favourable outcomes = 3 + 2 - 1 = 4.
Probability = 46 = 23.

Question 7

The odds in favour of an event are 3:5. Find the probability that the event occurs.

A

5/8✓✗

B

3/5✓✗

C

3/8✓✗

D

5/3✓✗

Explanation

Odds in favour of a:b means Probability = aa + b.
Here a = 3, b = 5, so Probability = 33 + 5 = 38.

Question 8

Two dice are rolled together. Find the probability that the sum of the numbers on them is 8.

A

5/36✓✗

B

1/6✓✗

C

7/36✓✗

D

1/9✓✗

Explanation

Total outcomes when 2 dice are rolled = 6 × 6 = 36.
Pairs whose sum is 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 favourable outcomes.
Probability = 536.

Question 9

Among 5 white and 7 black balls placed in a box, two are drawn one after another without replacement. Find the probability that both balls drawn are white.

A

7/33✓✗

B

5/22✓✗

C

5/33✓✗

D

7/22✓✗

Explanation

P(first ball white) = 512.
After removing one white ball, 4 white balls remain out of 11 total.
P(second ball white) = 411.
P(both white) = 512 × 411 = 20132 = 533.

Question 10

A committee of 3 members is to be selected at random from a group of 5 men and 4 women. Find the probability that the committee consists of exactly 2 men and 1 woman.

A

5/21✓✗

B

10/21✓✗

C

4/21✓✗

D

11/21✓✗

Explanation

Total ways to choose 3 people from 9 = C(9,3) = 84.
Favourable ways = choose 2 men from 5 and 1 woman from 4 = C(5,2) × C(4,1) = 10 × 4 = 40.
Probability = 4084 = 1021.

Question 11

In a class, 60% of the students are boys and the rest are girls. If 30% of the boys and 40% of the girls wear glasses, find the probability that a student selected at random wears glasses.

A

0.30✓✗

B

0.36✓✗

C

0.40✓✗

D

0.34✓✗

Explanation

P(boy and wears glasses) = 0.6 × 0.3 = 0.18.
P(girl and wears glasses) = 0.4 × 0.4 = 0.16.
P(wears glasses) = 0.18 + 0.16 = 0.34.

Question 12

A and B are two independent events such that P(A) = 0.4 and P(B) = 0.5. Find the probability that at least one of them occurs.

A

0.9✓✗

B

0.2✓✗

C

0.7✓✗

D

0.3✓✗

Explanation

P(A does not occur) = 1 - 0.4 = 0.6.
P(B does not occur) = 1 - 0.5 = 0.5.
P(neither occurs) = 0.6 × 0.5 = 0.3.
P(at least one occurs) = 1 - 0.3 = 0.7.

Shortcut Method

P(none) = q1 × q2 = 0.6 × 0.5 = 0.3
∴ P(at least one) = 1 - 0.3 = 0.7 ✓

Question 13

From a group of 6 boys and 4 girls, a team of 4 is selected at random. Find the probability that the team has no girls.

A

1/14✓✗

B

3/14✓✗

C

2/7✓✗

D

1/7✓✗

Explanation

Total ways to choose 4 from 10 people = C(10,4) = 210.
Ways with no girls = choose all 4 from the 6 boys = C(6,4) = 15.
Probability = 15210 = 114.

Question 14

Two letters are drawn at random, one after another without replacement, from the word COUNTRY. Find the probability that both letters are vowels.

A

2/21✓✗

B

1/7✓✗

C

1/21✓✗

D

5/21✓✗

Explanation

COUNTRY has 7 letters, of which 2 are vowels (O, U).
P(first letter a vowel) = 27.
After removing it, 1 vowel remains out of 6 letters.
P(second letter a vowel) = 16.
P(both vowels) = 27 × 16 = 242 = 121.

Question 15

A box contains 8 defective and 12 non-defective items. If 2 items are drawn at random without replacement, find the probability that both are non-defective.

A

32/95✓✗

B

34/95✓✗

C

33/95✓✗

D

3/5✓✗

Explanation

Total items = 8 + 12 = 20.
P(first item non-defective) = 1220.
After removing one non-defective item, 11 non-defective remain out of 19.
P(second item non-defective) = 1119.
P(both non-defective) = 1220 × 1119 = 132380 = 3395.

Question 16

In a single throw of two dice, find the probability of getting a total that is a prime number.

A

1/3✓✗

B

5/12✓✗

C

7/12✓✗

D

1/2✓✗

Explanation

Total outcomes = 36.
Sums that are prime (2, 3, 5, 7, 11) occur in 1 + 2 + 4 + 6 + 2 = 15 ways.
Probability = 1536 = 512.

Question 17

The probability that a certain machine will produce a defective item is 1/5. If two items are produced independently, find the probability that exactly one of them is defective.

A

4/25✓✗

B

1/25✓✗

C

16/25✓✗

D

8/25✓✗

Explanation

P(defective) = 15, P(not defective) = 45.
Exactly one defective can happen in 2 ways: (defective, ok) or (ok, defective).
P(exactly one) = 2 × 15 × 45 = 2 × 425 = 825.

Shortcut Method

Exactly one defective → 2pq
2 × 15 × 45 = 2 × 425
∴ P = 825 ✓

Question 18

Out of 6 red, 4 green and 5 blue balls kept in a box, three are drawn together at random. Find the probability that the three balls are all of different colours.

A

24/91✓✗

B

20/91✓✗

C

18/91✓✗

D

30/91✓✗

Explanation

Total ways to choose 3 balls from 15 = C(15,3) = 455.
Favourable ways (one of each colour) = 6 × 4 × 5 = 120.
Probability = 120455 = 2491.

Question 19

There are two bags. Bag A contains 3 red and 2 black balls, and Bag B contains 2 red and 4 black balls. A bag is chosen at random and a ball is drawn from it. Find the probability that the ball drawn is red.

A

1/2✓✗

B

3/10✓✗

C

7/15✓✗

D

1/6✓✗

Explanation

P(red) = P(Bag A chosen) × P(red | A) + P(Bag B chosen) × P(red | B)
= 12 × 35 + 12 × 26
= 310 + 16.
Using 30ths: 930 + 530 = 1430 = 715.

Shortcut Method

P(red) = avg of 35 and 26 (bags equally likely)
35 = 0.6, 26 = 0.333
Avg = 0.467
∴ P(red) = 715 ✓

Question 20

A number is selected at random from the first 30 natural numbers. Find the probability that it is a multiple of 3 or a multiple of 7.

A

14/30✓✗

B

12/30✓✗

C

13/30✓✗

D

11/30✓✗

Explanation

Multiples of 3 from 1 to 30 = 10.
Multiples of 7 from 1 to 30 = 4 (7, 14, 21, 28).
Multiples of both 3 and 7 (i.e. 21) = 1.
Favourable = 10 + 4 - 1 = 13.
Probability = 1330.

Question 21

Seven men and five women are available to form a committee of 4 members, chosen at random. Find the probability that the committee includes at least 3 women.

A

7/33✓✗

B

4/33✓✗

C

2/11✓✗

D

5/33✓✗

Explanation

Total ways to choose 4 from 12 people = C(12,4) = 495.
Exactly 3 women and 1 man: C(5,3) × C(7,1) = 10 × 7 = 70.
Exactly 4 women: C(5,4) = 5.
Favourable (at least 3 women) = 70 + 5 = 75.
Probability = 75495 = 533.

Question 22

Suppose two dice are thrown at the same time. What is the probability that the product of the two resulting numbers is a perfect square?

A

1/6✓✗

B

1/4✓✗

C

2/9✓✗

D

5/18✓✗

Explanation

List every pair (a,b) with a, b from 1 to 6 whose product is a perfect square:
Product 1: (1,1). Product 4: (1,4), (4,1), (2,2). Product 9: (3,3). Product 16: (4,4). Product 25: (5,5). Product 36: (6,6).
Total favourable pairs = 1 + 3 + 1 + 1 + 1 + 1 = 8.
Probability = 836 = 29.

Question 23

In a group of 12 people, 7 can speak English, 5 can speak French, and 2 can speak both languages. If a person is selected at random from the group, find the probability that the person speaks at least one of the two languages.

A

5/6✓✗

B

2/3✓✗

C

3/4✓✗

D

11/12✓✗

Explanation

Number speaking English or French = (Number speaking English) + (Number speaking French) - (Number speaking both), to avoid counting the overlap twice.
= 7 + 5 - 2 = 10.
Probability = 1012 = 56.

Shortcut Method

|E∪F| = |E| + |F| - |E∩F|
= 7 + 5 - 2 = 10
∴ P = 1012 = 56 ✓

Question 24

A fair coin is tossed 4 times. Find the probability of getting at least 3 heads.

A

1/4✓✗

B

3/8✓✗

C

5/16✓✗

D

1/2✓✗

Explanation

Total outcomes for 4 tosses = 2⁴ = 16.
Exactly 3 heads: C(4,3) = 4 ways.
Exactly 4 heads: C(4,4) = 1 way.
At least 3 heads = 4 + 1 = 5.
Probability = 516.

Shortcut Method

At least 3 heads in 4 tosses → C(4,3)+C(4,4)
= 4 + 1 = 5 → out of 2&sup4;=16
∴ P = 516 ✓

Question 25

Bulbs in a carton include 3 defective and 7 non-defective ones. They are tested one by one, without replacement, until a defective bulb turns up. Find the probability that exactly 3 bulbs need to be tested.

A

3/40✓✗

B

21/40✓✗

C

7/40✓✗

D

9/40✓✗

Explanation

P(1st bulb non-defective) = 710.
P(2nd bulb non-defective | 1st non-defective) = 69 (6 non-defective left out of 9).
P(3rd bulb defective) = 38 (3 defective left out of 8).
P(exactly 3rd bulb is defective) = 710 × 69 × 38 = 126720 = 740.