Quantitative Aptitude
Probability
1,860 Questions
Probability Questions
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$\displaystyle \frac { 5 }{ 54 } $
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$\displaystyle \frac { 7 }{ 54 } $
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$\displaystyle \frac { 5 }{ 34 } $
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$\displaystyle \frac { 7 }{ 34 } $
A
Correct answer
Explanation
Total outcomes = 6^3 = 216. We need strictly increasing sequences (a < b < c). The number of ways to choose 3 distinct numbers from 6 is 6C3 = 20. Each set of 3 numbers has only 1 way to be arranged in increasing order. Probability = 20/216 = 5/54.
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$\displaystyle^{ 8 }{ { C }_{ 3 } }\frac { { 5 }^{ 8 } }{ { 6 }^{ 8 } } $
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$\displaystyle^{ 7 }{ { C }_{ 2 } }\frac { { 5 }^{ 5 } }{ { 6 }^{ 8 } } $
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$\displaystyle^{ 7 }{ { C }_{ 2 } }\frac { { 5 }^{ 5 } }{ { 6 }^{ 7 } } $
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None of these
B
Correct answer
Explanation
For the third six to occur on the eighth throw, exactly two sixes must occur in the first seven throws, and the eighth throw must be a six. The probability of getting two sixes in seven throws is 7C2 * (1/6)^2 * (5/6)^5. Multiplying this by the probability of a six on the eighth throw (1/6) gives 7C2 * (5^5 / 6^8).
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$\displaystyle \frac {3n+1}{4n+2}$
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$\displaystyle \frac {3}{4}$
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$\displaystyle \frac {2}{3}$
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$\displaystyle \frac {1}{2}$
D
Correct answer
Explanation
In a fair die, odd numbers are {1, 3, 5} and even numbers are {2, 4, 6}. The probability of rolling an even number is 1/2. In (2n+1) trials, the probability of getting an odd number of successes in a binomial distribution with p=1/2 is always 1/2.
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$\displaystyle\frac{1}{36}$
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$\displaystyle\frac{1}{6}$
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$\displaystyle\frac{3}{36}$
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$\displaystyle\frac{2}{6}$
B
Correct answer
Explanation
Total outcomes for two dice = 36. Outcomes with sum 10 or more: (4,6), (5,5), (6,4), (5,6), (6,5), (6,6). There are 6 such outcomes. Probability = 6/36 = 1/6.
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$\displaystyle\frac{x}{11};\quad x = 1$
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$\displaystyle\frac{x}{12};\quad x = 7$
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$\displaystyle\frac{x}{13};\quad x = 9$
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None of these
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$(1)\quad \displaystyle\frac{1}{2}\ \ (2)\quad \displaystyle\frac{1}{3}$
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$(1)\quad \displaystyle\frac{2}{5}\ \ (2)\quad \displaystyle\frac{1}{3}$
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$(1)\quad \displaystyle\frac{2}{7}\ \ (2)\quad \displaystyle\frac{2}{9}$
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$(1)\quad \displaystyle\frac{1}{4}\ \ (2)\quad \displaystyle\frac{2}{5}$
D
Correct answer
Explanation
Total balls = 7+8+5 = 20. (1) P(White) = 5/20 = 1/4. (2) Neither red nor white means green. P(Green) = 8/20 = 2/5.
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$\displaystyle \frac {1}{(13)^5}$
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$\displaystyle \frac {((12)^5}{(13)^5}$
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$\displaystyle \frac {(13)^5-(12)^5}{(13)^5}$
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$\displaystyle \frac {1}{(12)^5}$
C
Correct answer
Explanation
With replacement, the probability of drawing an Ace on one draw is 1/13, so the probability of no Ace is 12/13. The probability of at least one Ace in five draws is therefore 1 - (12/13)^5, equal to ((13)^5 - (12)^5)/(13)^5.
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$\displaystyle \frac {7}{16}$
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$\displaystyle \frac {3}{4}$
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$\displaystyle \frac {9}{16}$
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$\displaystyle \frac {1}{2}$
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$P(A)=\displaystyle \frac {4}{17}; P(B)=\frac {1}{13}$
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$P(A)=\displaystyle \frac {1}{13}; P(B)=\frac {1}{13}$
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$P(A)=\displaystyle \frac {1}{13}; P(B)=\frac {1}{17}$
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$P(A)=\displaystyle \frac {16}{221}; P(B)=\frac {4}{51}$
B
Correct answer
Explanation
P(B) = 4/52 = 1/13. P(A) is the probability the second card is an ace. By symmetry, the probability of any specific card being an ace is 4/52 = 1/13. Thus P(A) = 1/13.
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$\displaystyle \frac {m}{m+n}$
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$\displaystyle \frac {m(n-1)}{(m+n)(m+n-1)}$
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$\displaystyle \frac {m(m-1)}{(m+n)(m+n-1)}$
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$\displaystyle \frac {mn}{(m+n)(m+n-1)}$
A
Correct answer
Explanation
By symmetry, the probability of any specific ball being drawn at any specific position (without knowing the outcomes of other draws) is equal to the initial proportion of that ball in the urn. Since there are m white balls out of m+n total balls, the probability that the second ball is white is m/(m+n).
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$\dfrac{9}{28}$
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$\dfrac{19}{28}$
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$\dfrac{9}{14}$
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$\dfrac{5}{14}$
A
Correct answer
Explanation
Total ways to choose 3 cards from 8 is C(8,3) = 56. To get 'ASH', we need one A (from 3), one S (from 2), and one H (from 3). Ways = 3 * 2 * 3 = 18. Probability = 18/56 = 9/28.
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$\dfrac{11}{17}$
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$\dfrac4{17}$
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$\dfrac6{17}$
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$\dfrac7{17}$
A
Correct answer
Explanation
Total number of balls = 4 red + 6 blue + 7 yellow = 17. The number of balls that are not blue is 4 red + 7 yellow = 11. The probability of selecting a non-blue ball is 11/17.
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$\displaystyle \frac{10}{49}$
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$\displaystyle \frac{13}{49}$
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$\displaystyle \frac{3}{49}$
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$\displaystyle \frac{1}{49}$
D
Correct answer
Explanation
Total cards = 52 - 3 = 49. The 10 of hearts is still in the deck. Probability = 1 / 49.
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${ \left( \cfrac { 1 }{ 5 } \right) }^{ 6 }\\ \\ $
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${ \left( \cfrac { 4 }{ 5 } \right) }^{ 6 }\\ \\ $
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$_{ 1 }^{ 6 }{ C }{ \left( \cfrac { 1 }{ 5 } \right) }^{ 5 }{ \left( \cfrac { 4 }{ 5 } \right) }^{ 1 }\\ \\ $
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$_{ 1 }^{ 6 }{ C }{ \left( \cfrac { 1 }{ 5 } \right) }^{ 1 }{ \left( \cfrac { 4 }{ 5 } \right) }^{ 5 }\\ \\ $
B
Correct answer
Explanation
This is a binomial distribution problem where n=6, p=0.2 (chance of disease), and q=0.8 (chance of no disease). The probability that no worker suffers from the disease is q^6 = (0.8)^6 = (4/5)^6.
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$\left( 200,\ \cfrac { 1 }{ 9 } ,\ \cfrac { 8 }{ 9 } \right) $
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$\left( 200,\ \cfrac { 2 }{ 9 } ,\ \cfrac { 7 }{ 9 } \right) $
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$\left( 200,\ \cfrac { 4 }{ 9 } ,\ \cfrac { 5 }{ 9 } \right) $
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$\left( 200,\ \cfrac { 1 }{ 4 } ,\ \cfrac { 3 }{ 4 } \right) $
A
Correct answer
Explanation
Sum of 9 with two dice: (3,6), (4,5), (5,4), (6,3) = 4 outcomes out of 36. Probability p = 4/36 = 1/9. q = 1 - p = 8/9. Binomial distribution is (n, p, q) = (200, 1/9, 8/9).