Probability · NDA previous year questions

For the next four (04) items that follow: Consider events A, B, C, D, E of the sample space $S=\{n:n\text{ is an integer such that }10\le n\le 20\}$ given by: A is the set of all even numbers. B is the set of all prime numbers. $C=\{15\}$. D is the set of all integers $\le 16$. E is the set of all double digit numbers expressible as a power of 2. C and E are

1. A.Mutually exclusive events but not elementary events
2. B.Exhaustive events but not mutually exclusive events
3. C.Mutually exclusive and exhaustive events
4. D.Elementary and mutually exclusive events

For any two events A and B, which one of the following holds?

1. A.$P(A\cap B)\le P(A)\le P(A\cup B)\le P(A)+P(B)$
2. B.$P(A\cup B)\le P(A)\le P(A\cap B)\le P(A)+P(B)$
3. C.$P(A\cup B)\le P(B)\le P(A\cap B)\le P(A)+P(B)$
4. D.$P(A\cap B)\le P(B)\le P(A)+P(B)\le P(A\cup B)$

The probability that in a random arrangement of the letters of the word ‘UNIVERSITY’, the two I’s do not come together is

1. A.4/5
2. B.1/5
3. C.1/10
4. D.9/10

There are 4 white and 3 black balls in a box. In another box, there are 3 white and 4 black balls. An unbiased dice is rolled. If it shows a number less than or equal to 3, then a ball is drawn from the second box, otherwise from the first box. If the ball drawn is black, then the probability that the ball was drawn from the first box is

1. A.1/2
2. B.6/7
3. C.4/7
4. D.3/7

Two students X and Y appeared in an examination. The probability that X will qualify the examination is 0.05 and Y will qualify the examination is 0.10. The probability that both will qualify the examination is 0.02. What is the probability that only one of them will qualify the examination?

1. A.0.15
2. B.0.14
3. C.0.12
4. D.0.11

A fair coin is tossed four times. What is the probability that at most three tails occur?

1. A.7/8
2. B.15/16
3. C.13/16
4. D.3/4

For the next four (04) items that follow: Consider events A, B, C, D, E of the sample space $S=\{n:n\text{ is an integer such that }10\le n\le 20\}$ given by: A is the set of all even numbers. B is the set of all prime numbers. $C=\{15\}$. D is the set of all integers $\le 16$. E is the set of all double digit numbers expressible as a power of 2. A, B and C are

1. A.Mutually exclusive events but not exhaustive events
2. B.Exhaustive events but not mutually exclusive events
3. C.Mutually exclusive and exhaustive events
4. D.Elementary events

What is $(1001{)}_2$ equal to ?

1. A.$(5{)}_{10}$
2. B.$(9{)}_{10}$
3. C.$(17{)}_{10}$
4. D.$(11{)}_{10}$

Let $X$ denote the number of scores which exceed 4 in 18 tosses of a symmetrical die. Consider the following statements: 1. The arithmetic mean of $X$ is 6. 2. The standard deviation of $X$ is 2. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

A box contains 3 white and 2 black balls. Two balls are drawn at random one after the other. If the balls are not replaced, what is the probability that both the balls are black?

1. A.2/5
2. B.1/5
3. C.1/10
4. D.None of the above

For the next four (04) items that follow: Number X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set $\{1,2,3,4,5,6,7\}$. Let $Z=(X+Y)$. What is $P(Z=5)$ equal to?

1. A.1/2
2. B.1/3
3. C.1/4
4. D.1/6

For the next four (04) items that follow: Number X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set $\{1,2,3,4,5,6,7\}$. Let $Z=(X+Y)$. What is $P(Z=5)$ equal to? What is $P(Z=10)$ equal to?

1. A.0
2. B.1/2
3. C.1/3
4. D.1/5

For the next four (04) items that follow: Number X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set $\{1,2,3,4,5,6,7\}$. Let $Z=(X+Y)$. What is $P(Z=5)$ equal to? What is $P(Z>11)$ equal to?

1. A.0
2. B.1/4
3. C.1/6
4. D.1/12

For the next four (04) items that follow: Number X is randomly selected from the set of odd numbers and Y is randomly selected from the set of even numbers of the set $\{1,2,3,4,5,6,7\}$. Let $Z=(X+Y)$. What is $P(Z=5)$ equal to? What is $P(Z\text{ is the product of two prime numbers})$ equal to?

1. A.0
2. B.1/2
3. C.1/4
4. D.None of the above

It has been found that if $A$ and $B$ play a game 12 times, $A$ wins 6 times, $B$ wins 4 times and they draw twice. $A$ and $B$ take part in a series of 3 games. The probability that they win alternately, is:

1. A.5/12
2. B.5/36
3. C.19/27
4. D.5/27

What is the sum of the two numbers $(11110{)}_2$ and $(1010{)}_2$?

1. A.$(101000{)}_2$
2. B.$(110000{)}_2$
3. C.$(100100{)}_2$
4. D.$(101100{)}_2$

What is the seventh term of the sequence $0,3,8,15,24,\dots$?

1. A.63
2. B.48
3. C.35
4. D.33

What is $\tan 15^{\circ }$ equal to?

1. A.$2-\sqrt{3}$
2. B.$2+\sqrt{3}$
3. C.$1-\sqrt{3}$
4. D.$1+\sqrt{3}$

Consider the following : 1. $\tan \left(\frac{\pi }{6}\right)$ 2. $\tan \left(\frac{3\pi }{4}\right)$ 3. $\tan \left(\frac{5\pi }{4}\right)$ 4. $\tan \left(\frac{2\pi }{3}\right)$ What is the correct order?

1. A.$1<4<2<3$
2. B.$4<2<1<3$
3. C.$4<2<3<1$
4. D.$1<4<3<2$

If $\cos x=\frac{1}{3}$, then what is $\sin x\cdot \cot x\cdot \mathrm{cosec}x\cdot \tan x$ equal to?

1. A.$\frac{2}{3}$
2. B.$\frac{3}{2}$
3. C.2
4. D.1

What is ${\log }_{81}243$ equal to?

1. A.0.75
2. B.1.25
3. C.1.5
4. D.3

If three events $A,B,C$ are mutually exclusive, then which one of the following is correct?

1. A.$P(A\cup B\cup C)=0$
2. B.$P(A\cup B\cup C)=1$
3. C.$P(A\cap B\cap C)=0$
4. D.$P(A\cap B\cap C)=1$

If $A$ and $B$ are independent events such that $P(A)=\frac{1}{5}$, $P(A\cup B)=\frac{7}{10}$, then what is $P(\overline{B})$ equal to?

1. A.$\frac{2}{7}$
2. B.$\frac{3}{7}$
3. C.$\frac{3}{8}$
4. D.$\frac{7}{9}$

In a binomial distribution, the occurrence and the non-occurrence of an event are equally likely and the mean is 6. The number of trials required is

1. A.15
2. B.12
3. C.10
4. D.6

A die is tossed twice. What is the probability of getting a sum of 10?

1. A.$\frac{1}{18}$
2. B.$\frac{1}{6}$
3. C.$\frac{1}{12}$
4. D.$\frac{5}{12}$