Probability · NDA previous year questions

Consider the following : 1. ${\mathrm{cosec}}^{-1}\left(-\frac{2}{\sqrt{3}}\right)=-\frac{\pi }{3}$ 2. ${\sec }^{-1}\left(\frac{2}{\sqrt{3}}\right)=\frac{\pi }{6}$ Which of the above is/are correct?

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

Consider the following statements with regard to correlation coefficient $r$ between random variables $x$ and $y$ : 1. $r=+1$ or $-1$ means there is a linear relation between $x$ and $y$. 2. $-1\le r\le 1$ and $r^2$ is a measure of the linear relationship between the variables. Which of the statements given above 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 6 distinct dolls. From this box, three dolls are randomly selected one by one with replacement. What is the probability of selecting 3 distinct dolls?

1. A.5/54
2. B.12/25
3. C.1/20
4. D.5/9

If $A$ and $B$ are events such that $P(A\cup B)=0\cdot 5$, $P(\overline{B})=0\cdot 8$ and $P(A\mid B)=0\cdot 4$, then what is $P(A\cap B)$ equal to ?

1. A.$0\cdot 08$
2. B.$0\cdot 02$
3. C.$0\cdot 8$
4. D.$0\cdot 2$

What is the probability of having $53$ Sundays or $53$ Mondays in a leap year?

1. A.$2/7$
2. B.$3/7$
3. C.$4/7$
4. D.$5/7$

Three digital numbers are formed using the digits $0,2,4,6,8$. A number is chosen at random out of these numbers. What is the probability that the number has the same digits?

1. A.$1/16$
2. B.$1/25$
3. C.$16/25$
4. D.$1/645$

A lot of $4$ white and $4$ red balls is randomly divided into two halves. What is the probability that there will be $2$ red and $2$ white balls in each half?

1. A.$18/35$
2. B.$3/35$
3. C.$1/2$
4. D.None of the above

Consider the following statements : If $A$ and $B$ are independent events, then 1. $A$ and $\bar{B}$ are independent 2. $\bar{A}$ and $B$ are independent 3. $\bar{A}$ and $\bar{B}$ are independent Which of the above statements is/are correct?

1. A.$3$ only
2. B.$1$ and $2$ only
3. C.$1,2$ and $3$
4. D.None of the statements $1,2$ and $3$ is correct

Consider the following statements: 1. The probability that there are 53 Sundays in a leap year is twice the probability that there are 53 Sundays in a non-leap year. 2. The probability that there are 5 Mondays in the month of March is thrice the probability that there are 5 Mondays in the month of April. Which of the statements given above is/are correct?

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

In tossing three coins at a time, what is the probability of getting at most one head?

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

Two balls are selected from a box containing 2 blue and 7 red balls. What is the probability that at least one ball is blue?

1. A.$\frac{2}{9}$
2. B.$\frac{7}{9}$
3. C.$\frac{5}{12}$
4. D.$\frac{7}{12}$

If the equation $x^2-bx+1=0$ does not possess real roots, then which one of the following is correct?

1. A.$-3<b<3$
2. B.$-2<b<2$
3. C.$b>2$
4. D.$b<-2$

The probability of guessing a correct answer is $\frac{x}{12}$. If the probability of not guessing the correct answer is $\frac{2}{3}$, then what is x equal to?

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

If the system of equations $2x+3y=7$ and $2ax+(a+b)y=28$ has infinitely many solutions, then which one of the following is correct?

1. A.$a=2b$
2. B.$b=2a$
3. C.$a=-2b$
4. D.$b=-2a$

An observed event $B$ can occur after one of the three events $A_1,A_2,A_3$. If $P(A_1)=P(A_2)=0.4$, $P(A_3)=0.2$ and $P(B\mid A_1)=0.25$, $P(B\mid A_2)=0.4$, $P(B\mid A_3)=0.125$, what is the probability of $A_1$ after observing $B$?

1. A.1/3
2. B.6/19
3. C.20/57
4. D.2/5

The probability distribution of random variable X with two missing probabilities $p_1$ and $p_2$ is given below: $$\begin{array}{ll}\text{X}& \text{P(X)}\\ 1& k\\ 2& p_1\\ 3& 4k\\ 4& p_2\\ 5& 2k\end{array}$$ It is further given that $P(X\le 2)=0.25$ and $P(X\ge 4)=0.35$. Consider the following statements: 1. $p_1=p_2$ 2. $p_1+p_2=P(X=3)$ Which of the statements given above is/are correct?

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

Three letters are randomly selected from the 26 capital letters of the English Alphabet. What is the probability that the letter 'A' will not be included in the choice ?

1. A.1/2
2. B.23/26
3. C.12/13
4. D.25/36

A coin is tossed 10 times. The number of heads minus the number of tails in 10 tosses is considered as the outcome of the experiment. What is the number of points in the sample space ?

1. A.10
2. B.11
3. C.21
4. D.99

Two numbers are successively drawn from the set $\{1,2,3,4,5,6,7\}$ without replacement and the outcomes recorded in that order. What is the number of elementary events in the random experiment ?

1. A.49
2. B.42
3. C.21
4. D.14

The probabilities of two events $A$ and $B$ are given as $P(A)=0.8$ and $P(B)=0.7$. What is the minimum value of $P(A\cap B)$ ?

1. A.0
2. B.0.1
3. C.0.5
4. D.1

Two numbers $X$ and $Y$ are simultaneously drawn from the set $\{1,2,3,4,5,6,7,8,9,10\}$. What is the conditional probability of exactly one of the two numbers $X$ and $Y$ being even, given $(X+Y)=15$ ?

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

If $A,B,C$ are any three arbitrary events, then which one of the following expressions shows that both $A$ and $B$ occur but not $C$?

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

By Bayes' theorem, which one of the following probabilities is calculated?

1. A.Prior probability
2. B.Likelihood probability
3. C.Posterior probability
4. D.Conditional probability

Given that $P(A)=1/3$, $P(B)=1/4$, $P(A\mid B)=1/6$, then what is $P(B\mid A)$ equal to?

1. A.$1/4$
2. B.$1/8$
3. C.$3/4$
4. D.$1/2$

If $A$ and $B$ are two mutually exclusive and exhaustive events with $P(B)=3P(A)$, then what is the value of $P(\overline{B})$?

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