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

If 4 whole numbers taken at random are multiplied together, then the chance that the last digit in the product is 1, 3, 7 or 9 is
  1. $\dfrac{16}{625}$
  2. $\dfrac{4}{125}$
  3. $\dfrac{8}{81}$

  4. none
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The last digit of a product of integers depends only on the last digits of the factors. For the product to end in 1, 3, 7, or 9, each of the 4 factors must be odd and not end in 5. There are 4 such digits (1, 3, 7, 9) out of 10 possible digits for each factor, so the probability is (4/10)^4 = (2/5)^4 = 16/625.

Multiple choice

You have been offered the chance to play a dice game in which you will receive $Rs. 20/-$ each time if the total of the points on the two dice is $6$, what will be your decision if it costs $Rs. 3/-$ per toss instead of $Rs. 2.50/-$

  1. Yes

  2. No

  3. Ambiguous

  4. Data Insufficient
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Probability of sum 6 with two dice: (1,5), (2,4), (3,3), (4,2), (5,1) = 5/36. Expected return per toss = (5/36) * 20 = 100/36 = 2.77. Since the cost is 3, the expected value is 2.77 - 3 = -0.23. Since the expected value is negative, the decision should be No.

Multiple choice

Three integers are selected simultaneously from the set of integers {1,2,3,4,5,.......50}. The probability that the selected numbers are consecutive, is equal to
  1. $\displaystyle \frac{9}{(25)(49)}$
  2. $\displaystyle \frac{6}{(25)(49)}$
  3. $\displaystyle \frac{3}{(25)(49)}$
  4. $\displaystyle \frac{1}{25(49)}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The total number of ways to select 3 integers from 50 is 50C3 = (50 * 49 * 48) / 6. The number of favorable outcomes where the three integers are consecutive is 48 (from {1,2,3} to {48,49,50}). The probability is therefore 48 / 50C3 = 48 / (50 * 49 * 8) = 6 / (50 * 49) = 3 / (25 * 49).

Multiple choice

There are 6 red and 5 black balls in a bag. Two balls are drawn at random one after another with replacement. The probability that 1st one is red and the second one is black is
  1. $\displaystyle \frac{30}{121}$
  2. $\displaystyle \frac{60}{121}$
  3. $\displaystyle \frac{36}{221}$
  4. $\displaystyle \frac{36}{121}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total balls = 11. Probability of red = 6/11. Probability of black = 5/11. Since it is with replacement, the events are independent. Probability = (6/11) * (5/11) = 30/121.

Multiple choice

An urn contains 5 white and 7 black balls. A second urn contains 7 white and 8 black balls. One ball is transfered from the 1st urn to the 2nd urn without noticing its colour. A ball is now drawn at random from the 2nd urn. The probability that it is white is
  1. $\displaystyle \frac{8}{92}$
  2. $\displaystyle \frac{89}{192}$
  3. $\displaystyle \frac{9}{92}$
  4. $\displaystyle \frac{98}{192}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Case 1: White ball transferred (5/12). Urn 2 now has 8 white, 8 black. P(W) = 8/16 = 1/2. Case 2: Black ball transferred (7/12). Urn 2 now has 7 white, 9 black. P(W) = 7/16. Total P(W) = (5/12 * 1/2) + (7/12 * 7/16) = 5/24 + 49/192 = 40/192 + 49/192 = 89/192.

Multiple choice

An urn A contains 8 black and 5 white balls. A second urn B contains 6 black and 7 white balls.A blind folded persons is asked to draw a ball selecting one of the urns, the probability that the ball drawn is white is
  1. $\displaystyle \frac{5}{13}$
  2. $\displaystyle \frac{6}{13}$
  3. $\displaystyle \frac{7}{13}$
  4. $\displaystyle \frac{9}{13}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Probability of choosing Urn A = 1/2. Probability of white from A = 5/13. Probability of choosing Urn B = 1/2. Probability of white from B = 7/13. Total probability = (1/2 * 5/13) + (1/2 * 7/13) = 1/2 * (12/13) = 6/13.

Multiple choice

A box contains 2 red, 3 blue and 4 black balls. $3$ balls are drawn at random from the box. $p$ is the probability for the balls to be of different colours; $q$ is the probability for $2$ balls of the same colour and the third of different colour; $r$ is the probability for all the $3$ balls to be of the same colour. Arrange them in the decreasing order of magnitude
  1. $p, q, r$
  2. $q, r, p$
  3. $q, p, r$
  4. $r, q, p$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total balls = 9. Total ways to draw 3 = 9C3 = 84. p (diff colors) = (2C1 * 3C1 * 4C1)/84 = 24/84 = 2/7 approx 0.28. r (same color) = (2C3 + 3C3 + 4C3)/84 = (0 + 1 + 4)/84 = 5/84 approx 0.06. q (2 same, 1 diff) = 1 - p - r = 1 - 24/84 - 5/84 = 55/84 approx 0.65. Thus q > p > r.

Multiple choice

A bag contains $6$ white and $4$ black balls. Another bag contains $4$ white and $6$ black balls. A die is rolled and if it shows a prime number a ball is taken from the 1st bag, otherwise a ball is taken from the second bag. The probability that the ball drawn is white is
  1. $\displaystyle \frac{1}{2}$
  2. $\displaystyle \frac{3}{5}$
  3. $\displaystyle \frac{2}{5}$
  4. $\displaystyle \frac{1}{4}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Probability of prime (2, 3, 5) = 3/6 = 1/2. If prime, pick from bag 1 (6/10 white). If not prime, pick from bag 2 (4/10 white). Total probability = (1/2 * 6/10) + (1/2 * 4/10) = 3/10 + 2/10 = 5/10 = 1/2.

Multiple choice

A bag X contains 4 white and 2 black balls.Another bag Y contains 3 white and 4 black balls.A bag is selected at random and a ball is drawnat random from it. I. the probability that the ball drawn is white is $\frac{17}{42}$. II. The probability that the ball drawn is black is $\frac{19}{42}$. Which of the above statements is correct

  1. only I

  2. only II

  3. both I and II

  4. neither I nor II
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

P(W) = P(Bag X)P(W|X) + P(Bag Y)P(W|Y) = 0.5 * (4/6) + 0.5 * (3/7) = 1/3 + 3/14 = (14+9)/42 = 23/42. P(B) = 1 - P(W) = 1 - 23/42 = 19/42. Statement II is correct.

Multiple choice

Two cards are drawn at random from $10$ cards numbered $1$ to $10$. The probability that their sum is even if both the cards are drawn together
  1. $\displaystyle \frac{1}{3}$
  2. $\displaystyle \frac{4}{9}$
  3. $\displaystyle \frac{5}{9}$
  4. $\displaystyle \frac{1}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Total ways to pick 2 cards from 10 is 10C2 = 45. To get an even sum, both cards must be even (5C2 = 10 ways) or both must be odd (5C2 = 10 ways). Total favorable = 20. Probability = 20/45 = 4/9.

Multiple choice

The probability of getting different suit cards and different denomination cards when two cards are drawn from a pack is
  1. $\displaystyle \frac{13}{17}$
  2. $\displaystyle \frac{13}{34}$
  3. $\displaystyle \frac{12}{17}$
  4. $\displaystyle \frac{6}{17}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Total ways to draw 2 cards from 52 is 52C2 = 1326. Ways to get same suit is 4 * 13C2 = 312. Ways to get same denomination is 13 * 4C2 = 78. Total same suit or same denomination = 312 + 78 = 390. Probability of same = 390/1326 = 5/17. Probability of different = 1 - 5/17 = 12/17.