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

In a bag there are 4 white, 4 red and 2 green balls. Two balls are drawn at random.What is the probability that at least one ball is of red color? एक बैग में 4 सफेद, 4 लाल और 2 हरे रंग की गेंद होती हैं। दो गेंदें यादृच्छिक रूप से चुनी जाती हैं। क्या प्रायिकता है कि कम से कम एक गेंद लाल रंग की हो?

  1. 1/3

  2. 3/4

  3. 3/5

  4. 2/3

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

Total balls = 4 + 4 + 2 = 10. Drawing 2 balls at random gives C(10,2) = 45 total outcomes. 'At least one red' means 1 red OR 2 reds. Cases: (i) 1 red, 1 non-red: C(4,1)×C(6,1) = 4×6 = 24. (ii) 2 reds: C(4,2) = 6. Total favorable = 24 + 6 = 30. Probability = 30/45 = 2/3. Alternatively, P(at least 1 red) = 1 - P(no red) = 1 - C(6,2)/C(10,2) = 1 - 15/45 = 30/45 = 2/3.

Multiple choice

Five coins are tossed simultaneously. What is the probability of getting more than 3 heads? पांच सिक्कों को एक साथ उछाला गया। क्या प्रायिकता है कि तीन से अधिक शीर्ष प्राप्त होंगे?

  1. 1/4

  2. 1/2

  3. 3/16

  4. 5/16

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

Five coins tossed: total outcomes = 2^5 = 32. More than 3 heads means 4 or 5 heads. Number of ways to get exactly 4 heads = C(5,4) = 5. Number of ways to get exactly 5 heads = C(5,5) = 1. Total favorable = 6. Probability = 6/32 = 3/16. The claimed answer C (3/16) is correct.

Multiple choice

Seven coins are tossed simultaneously. Find the probability of getting exactly 4 heads. सात सिक्कों को एक साथ उछाला जाता है। 4 शीर्ष प्राप्त होने की प्रायिकता ज्ञात कीजिये।

  1. 17/64

  2. 35/128

  3. 9/32

  4. 3/8

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

For 7 coin tosses, P(exactly 4 heads) = C(7,4) × (1/2)⁴ × (1/2)³ = 35 × 1/16 × 1/8 = 35/128. Here C(7,4) = 7!/(4!3!) = 35 represents ways to choose 4 heads from 7 tosses. Each specific outcome has probability (1/2)⁷ = 1/128, so total probability = 35/128.

Multiple choice

In a box, there are 10 black, 13 white and 12 red balls. One ball is picked up randomly. What is the probability that it is neither white nor red? एक बॉक्स में, वहाँ 10 काली, 13 सफेद और 12 लाल गेंदे हैं। एक गेंद बेतरतीब ढंग से उठाई जाती है। क्या संभावना है कि यह न तो सफेद और न ही लाल हो?

  1. 2/7

  2. 5/7

  3. 1/7

  4. 3/7

  5. 4/7
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The question asks for the probability that a randomly picked ball is neither white nor red, which means it must be black. There are 10 black balls out of a total of 10 + 13 + 12 = 35 balls. Therefore, the probability of picking a black ball is 10/35 = 2/7. This is a straightforward probability calculation.

Multiple choice

A bag contains 20 tickets, numbered from 1 to 20. A ticket is drawn and then another ticket is drawn without replacement. The probability that both the tickets will show odd numbers is एक बैग में 20 टिकट हैं, 1 से 20 तक हैं। एक टिकट निकाला जाता है और फिर दूसरा टिकट बिना विस्थापन के निकाला जाता है। इस बात की क्या प्रायिकता है कि दोनों टिकट विषम संख्या दर्शाते हैं?

  1. 5/19

  2. 4/19

  3. 9/19

  4. 11/38

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

There are 10 odd numbers (1, 3, 5, 7, 9, 11, 13, 15, 17, 19). Probability of first odd = 10/20 = 1/2. After drawing one odd, 9 odd remain out of 19 tickets. Probability = (10/20) × (9/19) = 1/2 × 9/19 = 9/38. This is not among options A-D, so E is correct.

Multiple choice

Answer the following questions based on the statement given below: Two dice are rolled simultaneously. Find the probability of getting a sum greater than 9.
  1. $\(\frac{1}{2}\)$
  2. $\(\frac{1}{3}\)$
  3. $\(\frac{1}{4}\)$
  4. $\(\frac{1}{5}\)$
  5. $\(\frac{1}{6}\)$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

When rolling two dice, total outcomes = 6 × 6 = 36. Sums greater than 9 are: 10 (4,6; 5,5; 6,4), 11 (5,6; 6,5), 12 (6,6) — totaling 6 favorable outcomes. Probability = 6/36 = 1/6.

Multiple choice

Four coins are tossed together. Find the probability of getting exactly 3 heads. चार सिक्के एक साथ उछाले जाते हैं। 3 शीर्ष प्राप्त होने की संभावना ज्ञात कीजिये।

  1. 3/8

  2. 1/8

  3. 1/4

  4. 5/16

  5. None of these / इनमें से कोई नहीं
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total outcomes when tossing 4 coins = 2⁴ = 16. Favorable outcomes (exactly 3 heads) = C(4,3) = 4. The 4 outcomes are: HHHT, HHTH, HTHH, THHH. Probability = 4/16 = 1/4.

Multiple choice

A bag contains 12 Green and 8 White balls. Four balls are drawn out one by one and not replaced. What is the probability that they are in alternate colours?

  1. 308/4845

  2. 317/4845

  3. 414/4845

  4. 407/4845

  5. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Two alternating color patterns possible: G-W-G-W or W-G-W-G. For G-W-G-W: (12/20)(8/19)(11/18)(7/17). For W-G-W-G: (8/20)(12/19)(7/18)(11/17). Notice numerators are same (12*8*11*7). Total = 2*(12*8*11*7)/(20*19*18*17) = 2*7392/116280 = 14784/116280 = 317/4845. B matches this.

Multiple choice

Four coins are tossed simultaneously. Find the probability of getting exactly 3 tails. चार सिक्कों को एक साथ उछाला जाता हैं । ठीक 3 पुच्छ प्राप्त होने की संभावना ज्ञात कीजिये।

  1. 1/2

  2. 1/8

  3. 1/4

  4. 3/4

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

Total outcomes when tossing 4 coins = 2^4 = 16. Favorable outcomes with exactly 3 tails: TTTH, TTHT, THTT, HTTT = 4 outcomes. Probability = 4/16 = 1/4. This is option C.

Multiple choice

A box contains 10 different balls. Five balls are drawn simultaneously and then replaced and then seven balls are drawn. Then the probability that exactly three balls are common to the two draws is (p), then the value of (p) is:
  1. $\(\frac{1}{12}\)$
  2. $\(\frac{3}{12}\)$
  3. $\(\frac{5}{12}\)$
  4. $\(\frac{7}{12}\)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We need exactly 3 common balls between two draws. First draw selects 5 balls from 10. Second draw selects 7 balls from 10. Probability = [C(5,3) × C(5,4)] / C(10,7) = (10 × 5) / 120 = 50/120 = 5/12. The numerator counts ways to choose 3 common from first 5 and 4 new from remaining 5.

Multiple choice

A bag contains 4 pink and 5 yellow balls. One ball is drawn randomly. What is the probability that the ball drawn is Pink? एक बैग में 4 गुलाबी और 5 पीली गेंदें हैं। एक गेंद यादृच्छया निकाली जाती है। निकाली गई गेंद के गुलाबी होने की प्रायिकता क्या है?

  1. 4/9

  2. 2/9

  3. 1/9

  4. 7/9

  5. none of these इसमें से कोई नही
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total balls = 4 pink + 5 yellow = 9. Probability of drawing a pink ball = number of pink balls / total balls = 4/9. Option B (2/9) is incorrect as it only counts half the pink balls, and Option D (7/9) would represent the probability of drawing either pink OR yellow.

Multiple choice

There are 3 green, 4 orange and 5 white color balls in a bag. If a balls is picked at random, what is the probability of having either a orange or a white balls? एक बैग में 3 हरे, 4 नारंगी और 5 सफेद रंग के बॉल हैं। यदि एक बॉल यादृच्छया चुना जाता है, तो नारंगी या सफेद बॉल होने की प्रायिकता क्या है?

  1. 3/4

  2. 5/9

  3. 1/4

  4. 6/7

  5. 3/5
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total balls = 3 green + 4 orange + 5 white = 12. Favorable outcomes (orange or white) = 4 + 5 = 9. Probability = favorable/total = 9/12 = 3/4.

Multiple choice

A box contains 4 white balls, 5 black balls and 7 green balls. In how many ways can 4 balls be drawn from the box, if at least one white ball, one black ball and one green ball is to be included in a draw? एक डिब्बे में 4 सफेद गेंदें, 5 काली गेंदें और 7 हरी गेंदें हैं। यदि ड्रॉ में कम से कम एक सफेद गेंद, एक काली गेंद और एक हरी गेंद शामिल की जाती है, तो बॉक्स से 4 गेंदें कितने तरीकों से निकाली जा सकती हैं?

  1. 900

  2. 870

  3. 1020

  4. 910

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

Total ways to draw 4 balls from 16 = 16C4 = 1820. Subtract cases where at least one color is missing. No white: 12C4 = 495. No black: 11C4 = 330. No green: 9C4 = 126. But we subtracted cases where two colors are missing twice, so add back: 8C4 (no white, no black) + 7C4 (no white, no green) + 5C4 (no black, no green) = 70+35+5 = 110. Result = 1820 - (495+330+126) + 110 = 910. Alternatively, count directly: (2,1,1) distributions: 4C2×5C1×7C1 + 4C1×5C2×7C1 + 4C1×5C1×7C2 = 6×5×7 + 4×10×7 + 4×5×21 = 210+280+420 = 910. Option A (900) is close but incorrect.