Quantitative Aptitude

Probability

1,860 Questions
All Sample Space Setup 558 Permutation & Combination Counting 288 Multiplication Theorem (AND) 234 Addition Theorem (OR) 151 At Least One Probability 134 Complement Rule (P(Not A)) 130 Drawing Without Replacement 126 Conditional Probability 75 Bayes' Theorem 69 Drawing With Replacement 49 Success and Failure Rates 46

Probability Questions

Multiple choice

There are two bags containing white and black balls. In the first bag, there are 8 white and 6 black balls and in the second bag, there are 4 white and 7 black balls. One balls is drawn at random from any of these two bags. Find the probability of ball being black.

  1. 5/9

  2. 7/9

  3. 41/77

  4. 7/17
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using total probability: each bag is equally likely to be chosen (probability 1/2). Black probability from first bag = 6/14 = 3/7. Black probability from second bag = 7/11. Combined probability = (1/2)(3/7) + (1/2)(7/11) = 41/77.

Multiple choice

Bag 1 contains 2 white and 4 black balls. Bag 2 contains 5 white and 3 black balls. A ball is drawn from each bag, find the probability that both are black?
  1. $5/24$
  2. $1/4$
  3. $1/8$
  4. $1/2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Probability of both balls being black = P(black from bag 1) × P(black from bag 2). Bag 1 has 4 black out of 6 total (2/3). Bag 2 has 3 black out of 8 total (3/8). Product = (2/3) × (3/8) = 6/24 = 1/4. These are independent events since draws are from different bags.

Multiple choice

Each question contains a statement followed by Quantity I and Quantity II. Read the contents clearly and answer your questions accordingly. \begin{enumerate} \item A bag contains 7 black pens, 5 pink pens and 4 yellow pens. \item Quantity I: If 3 pens are drawn randomly, then find the probability of getting at least one pink pen? \item Quantity II: If 2 pens are drawn randomly, then find the probability of getting both the pens is either black or yellow?\end{enumerate}

  1. Quantity II > Quantity I

  2. Quantity I ≥ Quantity II

  3. Quantity I > Quantity II

  4. Quantity II ≥ Quantity I

  5. Quantity I = Quantity II or Relation cannot be established
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Quantity I: P(at least 1 pink) = 1 - P(no pink) = 1 - (11C3/16C3) = 1 - 165/560 = 395/560 = 79/112 ≈ 0.705. Quantity II: P(both black or yellow) = 11C2/16C2 = 55/120 = 11/24 ≈ 0.458. Cross-multiplying: 79*24 = 1896, 112*11 = 1232. Since 1896 > 1232, Quantity I (79/112) > Quantity II (11/24). Option C is correct.

Multiple choice

Which information are necessary to answer the questions asked- A bag contains balls of only three different colours i.e., red, yellow and green. 3 balls are drawn randomly. What is the probability that the balls drawn are of three different colours? A. The no. of yellow balls is two more than the no. of red balls. B. The Sum of the no. of yellow and green balls is three times the no. of red balls. C. The ratio of the no. of red balls to that of green balls is 3 : 4.

  1. A and either B or C

  2. Any two of them

  3. Only A and C together

  4. Question can’t be answered even after using all the informations.

  5. All statements are required
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Let r, y, g be numbers of red, yellow, green balls. A: y = r + 2. B: y + g = 3r. C: r/g = 3/4. Using A and C: g = 4r/3, then from A: y = r + 2. Need B to get actual values: r+2+4r/3 = 3r, giving r=6, y=8, g=8. Total = 22. P(3 different) = (6×8×8)/(22×21×20) = 384/9240 = 8/175. All three statements are required.

Multiple choice

In the given questions, two quantities are given, one as ‘Quantity I’ and another as ‘Quantity II’. You have to determine the relationship between two quantities and choose the appropriate option: Quantity I: A bag contains 50 balls which are pink, violet and red. Find the number of violet balls in the bag, if the probability of picking a pink ball is 3/5 and that of either a pink or violet ball is 4/5. Quantity II: A bag contains 40 balls pink, violet and red. The probability of picking a violet ball is 3/8. If the first ball was violet and without replacement, the probability of picking a pink ball is 4/13. Find the number of red balls.

  1. Quantity I < Quantity II

  2. Quantity I > Quantity II

  3. Quantity I ≤ Quantity II

  4. Quantity I ≥ Quantity II

  5. Quantity I = Quantity II or relationship cannot be established
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Quantity I: P(pink) = 3/5 means pink = 30. P(pink or violet) = 4/5 means pink+violet = 40. So violet = 10. Quantity II: P(violet) = 3/8 means violet = 15. After removing 1 violet, pink = 16 (since 16/39 = 4/13 after removing violet). Total = 40, so pink + violet + red = 40, giving 16 + 15 + red = 40, so red = 9. So Quantity I = 10, Quantity II = 9. Answer A is correct.

Multiple choice

Which information are necessary to answer the questions asked-| पूछे गए प्रश्नों के उत्तर देने के लिए कौन सी जानकारी आवश्यक है - A bag contains balls of only three different colours i.e., red, yellow and green. 3 balls are drawn randomly. What is the probability that the balls drawn are of three different colours? एक बैग में केवल तीन अलग-अलग रंगों की गेंदें हैं यानी लाल, पीला और हरा । 3 गेंदें बेतरतीब ढंग से निकाली जाती हैं। क्या संभावना है कि निकाली गई गेंदें तीन अलग-अलग रंगों की हैं? A. The no. of yellow balls is two more than the no. of red balls. B. The Sum of the no. of yellow and green balls is three times the no. of red balls. C. The ratio of the no. of red balls to that of green balls is 3 : 4. A. पीले रंग की गेंदों की संख्या लाल गेंदों से दो अधिक है। B. पीले और हरे रंग की गेंदों की संख्या का योग लाल गेंदों का तीन गुना है। C. लाल गेंदों का हरे रंग की गेंदों की संख्या से अनुपात 3: 4 है।

  1. A and either B or C A और B या C

  2. Any two of them उनमें से कोई भी दो

  3. Only A and C together केवल A और C एक साथ

  4. Question can’t be answered even after using all the informations. सभी कथनों का उपयोग करने के बाद भी प्रश्न का उत्तर नहीं दिया जा सकता है।

  5. All statements are required सभी कथनों की आवश्यकता है
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Let r, y, g be the counts of red, yellow, green balls. Total balls = r + y + g. We need P(drawing 3 different colors) = (r×y×g) / C(total,3) × 6. To find this, we need exact values or ratios. Statement A: y = r + 2. Statement B: y + g = 3r. Statement C: r:g = 3:4, so g = 4r/3. Using all three: from C, g = 4r/3. From A, y = r + 2. From B: (r+2) + 4r/3 = 3r, giving 7r/3 + 2 = 3r, so 2r/3 = 2, r = 3. Then y = 5, g = 4. Total = 12. P = (3×5×4)/C(12,3) × 6 = 60/220 × 6 = 18/22. Any two statements alone leave a variable undetermined. All three are needed.

Multiple choice

Each question contains a statement followed by Quantity I and Quantity II. Read the contents clearly and answer your questions accordingly. प्रत्येक प्रश्न में एक कथन होता है जिसके बाद मात्रा I और मात्रा II दी गई है। कथन को स्पष्ट रूप से पढ़ें और उस पर आधारित प्रश्नों का उत्तर दें। A bag contains 7 black pens, 5 pink pens and 4 yellow pens. Quantity I: If 3 pens are drawn randomly, then find the probability of getting at least one pink pen? Quantity II: If 2 pens are drawn randomly, then find the probability of getting both the pens is either black or yellow? एक बैग में 7 काली पेन, 5 गुलाबी पेन और 4 पीली पेन हैं। मात्रा I: यदि 3 पेन को अनियमित रूप से चुनते हैं, तो कम से कम एक गुलाबी पेन होने की क्या संभावना होगी? मात्रा II: यदि 2 पेन अनियमित रूप से चुनते हैं, तो दोनों पेन के काले या पीले पेन होने की क्या संभावना होगी?

  1. Quantity II > Quantity I मात्रा II > मात्रा I

  2. Quantity I ≥ Quantity II मात्रा I ≥ मात्रा II

  3. Quantity I > Quantity II मात्रा I > मात्रा II

  4. Quantity II ≥ Quantity I मात्रा II ≥ मात्रा I

  5. Quantity I = Quantity II or Relation cannot be established संबंध स्थापित नहीं किया जा सकता या मात्रा I = मात्रा II
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total pens = 7 + 5 + 4 = 16. Quantity I: P(at least one pink) = 1 - P(no pink) = 1 - C(11,3)/C(16,3) = 1 - (165/560) = 395/560 = 79/112 ≈ 0.705. Quantity II: P(both black or yellow) = [C(7,2) + C(4,2)]/C(16,2) = (21 + 6)/120 = 27/120 = 9/40 = 0.225. Comparing: Quantity I (0.705) > Quantity II (0.225). The answer correctly identifies this relationship.

Multiple choice

The questions given below contain two statements giving certain data. You have to decide whether the data given in the statements are sufficient for answering the question. What is the probability that at least two are green balls ? I. A basket contains 3 blue, 2 green and 5 red balls. II. The balls are picked at random. Mark answer - नीचे दिए गए प्रश्नों में दो कथनों के माध्यम से कुछ जानकारियां दी गई हैं आपको यह निर्णय करना है कि क्या कथनो में दी गई जानकारी उत्तर देने के लिए पर्याप्त हैं। कम से कम दो गेंदे हरी हो इसकी क्या प्रायकिता होगी ? (I) एक टोकरी में 3 नीली, 2 हरी तथा 5 लाल गेदें है। (II) गेदों को बेतरतीब निकाला गया।

  1. If statement I alone is sufficient but statement II alone is not sufficient. / यदि केवल कथन I ही पर्याप्त है लेकिन केवल कथन II नहीं

  2. If statement II alone is sufficient but statement I alone is not sufficient. / यदि केवल कथन II ही पर्याप्त है लेकिन केवल कथन I नहीं

  3. If each statement alone (either I or II) is sufficient./ यदि या केवल I या केवल II पर्याप्त हैं।

  4. If statement I and II together are not sufficient. / यदि I व II दोनों अपर्याप्त हैं।

  5. If both statement together are sufficient but neither statement alone is sufficient. / यदि दोनों कथन I व II एक साथ पर्याप्त हैं लेकिन अलग-अलग नहीं
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To find P(at least 2 green balls), we need to know how many balls are being picked. Statement I gives the composition: 3 blue, 2 green, 5 red (total 10 balls). Statement II says balls are picked randomly. But neither (nor both) statements specify HOW MANY balls are picked (2? 3? 5?). Without knowing the sample size, the probability cannot be calculated. The statements together are insufficient.

Multiple choice

Given below are two quantities named I and II. Based on the given information, you have to determine the relation between the two quantities. You should use the given data and your knowledge of Mathematics to choose among the possible answers. Quantity I: 5 balls are drawn at random from a bag which contains 10 white, 7 blue and 13 red balls. What is the probability if two balls are red? Quantity II: 7 marbles are drawn at random from a jar which contains 8 brown marbles, 6 blue marbles and 7 grey marbles. Find the probability if at least 4 marbles are brown.

  1. Quantity I > Quantity II

  2. Quantity I < Quantity II

  3. Quantity I ≥ Quantity II

  4. Quantity I ≤ Quantity II

  5. Quantity I = Quantity II or No relation
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Quantity I: Probability of exactly 2 red out of 5 from 10W, 7B, 13R (30 total) = C(13,2)×C(17,3)/C(30,5) = (78 × 680)/142506 ≈ 0.372. Quantity II: Probability of at least 4 brown out of 7 from 8 brown, 6 blue, 7 grey (21 total). This is complex to calculate but involves finding P(4 brown) + P(5 brown) + P(6 brown) + P(7 brown). The probability of getting 4 or more from only 8 browns in a sample of 7 from 21 total is relatively small. 0.372 > small probability, so Quantity I > Quantity II.

Multiple choice

Find the correct relationship between given quantities. दी गई मात्राओं के बीच सही संबंध ज्ञात कीजिये। Quantity I- A bag contains 3 green, 2 yellow and 3 purple balls. Two balls are drawn at random, what is the probability that no ball is purple. Quantity II- A bag contain 4 red balls and 6 yellow balls.If three balls are picked randomly then what is the probability that they are of the same color. मात्रा I- एक बैग में 3 हरे, 2 पीले और 3 बैंगनी गेंदे हैं। दो गेंदों को यादृच्छिक रूप से निकला जाता है, कोई गेंद बैंगनी नहीं होने की प्रायिकता ज्ञात कीजिये ? मात्रा II- एक बैग में 4 लाल और 6 पीली गेंदे हैं। यदि तीन गेंदों को अनियमित रूप से निकला जाता है तो एक ही रंग के होने की प्रायिकता क्या होगी ?

  1. Quantity I > Quantity II मात्रा I> मात्रा II

  2. Quantity I ≥ Quantity II मात्रा I ≥मात्रा II

  3. Quantity II > Quantity I मात्रा II> मात्रा I

  4. Quantity II ≥ Quantity I मात्रा II≥ मात्रा I

  5. Quantity I = Quantity II or Relation cannot be established मात्रा I = मात्रा II या संबंध स्थापित नहीं किया जा सकता है
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Quantity I: Total balls = 3+2+3 = 8. Ways to draw 2 balls with no purple = C(5,2) = 10. Total ways = C(8,2) = 28. Probability = 10/28 = 5/14 ≈ 0.357. Quantity II: Total balls = 4+6 = 10. Ways to draw 3 same color: all red = C(4,3) = 4, all yellow = C(6,3) = 20, total = 24. Total ways = C(10,3) = 120. Probability = 24/120 = 1/5 = 0.2. Since 0.357 > 0.2, Quantity I > Quantity II.

Multiple choice

The questions given below contain two statements giving certain data. You have to decide whether the data given in the statements are sufficient for answering the question. Mark answer What is the probability that at least two are green balls? I. A basket contains 3 blue, 2 green and 5 red balls. II. The balls are picked at random.

  1. If statement I alone is sufficient but statement II alone is not sufficient.

  2. If statement II alone is sufficient but statement I alone is no sufficient.

  3. If each statement alone (either I or II) is sufficient.

  4. If statement I and II together are not sufficient.

  5. If both statement together are sufficient but neither statement alone is sufficient.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To find the probability that at least two are green balls, we need to know how many balls are being picked. Statement I gives the composition of the basket (10 balls total: 3 blue, 2 green, 5 red). Statement II only tells us balls are picked randomly, but doesn't specify the sample size. Without knowing how many balls are drawn, we cannot calculate any probability. Even combining both statements, the sample size is missing, making the data insufficient.

Multiple choice

Each question below contains a statement followed by Quantity I and Quantity II.You have to find the correct relationship. नीचे दिए गए प्रत्येक प्रश्न में मात्रा I और मात्रा II के बाद कथन दिया गया है। आपको सही सम्बन्ध स्थापित करना है। A box contains 12 black balls, 15 green balls, 13 yellow balls and 16 purple balls. एक बॉक्स में 12 काली गेंदें, 15 हरी गेंदें, 13 पीली गेंदें और 16 बैंगनी गेंदें हैं। Quantity I: If three balls are drawn at random then the probability that all the balls are either green or yellow. Quantity II: If three balls are drawn at random then the probability that all are of different colours. मात्रा I: यदि तीन गेंदों को यादृच्छिक रूप से निकला जाता है तो सभी गेंदें हरे या पीले रंग की होने की क्या संभावना हैं। मात्रा II: यदि तीन गेंदों को यादृच्छिक रूप से निकला जाता है तो सभी अलग-अलग रंगों के होने की क्या संभावना है।

  1. Quantity I ≤ Quantity II मात्रा I ≤ मात्रा II

  2. Quantity I ≥ Quantity II मात्रा I ≥ मात्रा II

  3. Quantity I > Quantity II मात्रा I > मात्रा

  4. Quantity I < Quantity II मात्रा I < मात्रा

  5. Quantity I = Quantity II, or relation cannot be established मात्रा I = मात्रा II, या संबंध स्थापित नहीं किया जा सकता है
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Total balls = 12+15+13+16 = 56. Quantity I: (15C3 + 13C3)/56C3 = (455 + 286)/34220 = 741/34220 ≈ 0.0217. Quantity II: Total ways = 56C3 = 34220. Favorable ways = (12×15×13)×6 = 14040. Probability = 14040/34220 ≈ 0.4103. Therefore Quantity I (0.0217) < Quantity II (0.4103).

Multiple choice

In bag A there are 5 red balls, X green balls and 7 yellow balls. Probability of drawing one green ball from bag A is 2/5. In bag B there are (X-3) red balls, (X-4) green balls and 6 yellow balls. 2 balls are drawn from bag B. Find the probability that both the balls are red colour? बैग A में 5 लाल गेंदें, X हरी गेंदें और 7 पीले रंग की गेंदें हैं। बैग A से एक हरे रंग के निकाले जाने की संभावना 2/5 है। बैग B में (X -3) लाल गेंदें, (X -4) हरी गेंदें और 6 पीले रंग की गेंदें हैं। बैग B से 2 गेंद निकाली जाती हैं। संभावना ज्ञात कीजिये कि दोनों गेंद लाल रंग की हों?

  1. 2/23

  2. 3/21

  3. 4/21

  4. 2/21

  5. None of these इनमे से कोई नहीं
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Total balls in Bag A = 5 + X + 7. P(green from Bag A) = X/(5+X+7) = 2/5. Solving: 5X = 2(12+X) => 5X = 24 + 2X => 3X = 24 => X = 8. Bag B has: (8-3)=5 red, (8-4)=4 green, 6 yellow = 15 total. P(2 red) = C(5,2)/C(15,2) = (5×4/2)/(15×14/2) = 10/105 = 2/21.

Multiple choice

Each question below contains a statement followed by Quantity I and Quantity II. Find both to find the relationship between them. Mark your answer accordingly. नीचे दिए गए प्रत्येक प्रश्न में मात्रा I और मात्रा II के बाद कथन दिए गए है। दोनों के बीच संबंध ज्ञात करने के लिए दोनों का मान ज्ञात कीजिये। तदनुसार सही उत्तर चुनिए। Quantity I: A bag contains 3 red, 4 green and 2 blue balls. Two balls are drawn at random. Find the probability that none of the ball drawn is blue. Quantity II: Fraction of work completed by A in 7 days if he is 20% more efficient than B who can complete the work in 12 days. मात्रा I: एक बैग में 3 लाल, 4 हरे और 2 नीली गेंदें हैं।यादृच्छिक रूप से दो गेंदें निकाली जाती हैं। संभावना ज्ञात कीजिये कि निकाली गई गेंदों में से कोई भी नीली नहीं है। मात्रा II: 7 दिनों में A द्वारा पूरा किए गए कार्य का भाग, यदि वह B से 20% अधिक कार्य कुशल है जो 12 दिनों में कार्य को पूरा कर सकता है।

  1. Q1 > Q2

  2. Q1 ≥ Q2

  3. Q1< Q2

  4. Q1 ≤ Q2

  5. Q1 = Q2 or relation cannot be established. Q1 = Q2 या सम्बन्ध स्थापित नहीं किया जा सकता है
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Quantity I: Total balls = 3 + 4 + 2 = 9. Non-blue balls = 7. Probability of drawing two non-blue balls = (7/9) × (6/8) = 7/12 ≈ 0.583. Quantity II: B completes work in 12 days, so B's rate = 1/12. A is 20% more efficient, so A's rate = 1.2 × (1/12) = 1/10. In 7 days, A completes 7 × (1/10) = 7/10 = 0.7 of work. Comparing: 0.583 < 0.7, so Q1 < Q2.