Mutually Exclusive Events

Q.

A rifleman is firing at a distant target and has only $10\%$ chance of hitting it. The minimum number of round he must fire in order to have $50\%$chance of hitting it at least once i

1. $7$

2. $8$

3. $9$

4. $6$

Q. Two events A and B have probabilities 0.25 and 0.50 respectively. The probability that both A and B occur simultaneously is 0.12. Then the probability that neither A nor B occurs is:

1. 0.13
2. 0.37
3. 0.63
4. 0.38

Q. A player tosses a coin and scores one point for every head and two point for every tail that truns up. He plays on until his scores reaches or psses $n$. $P_n$ denotes the probability of getting a scores of exactly $n$
$$\begin{array}{cc}\text{List I}& \text{List II}\\ \begin{array}{cc}\text{(a)}& \text{the value of}{P}_n\text{is}\end{array}& \text{(p)}1\\ \begin{array}{c}\text{(b) the value of}{P}_n+\frac{1}{2}{P}_{n-1}\end{array}& \text{(q)}\frac{5}{4}\\ \begin{array}{c}\text{(c)}2P_{101}+P_{100}\end{array}& \text{(r)}2\\ \begin{array}{c}\text{(d)}{P}_1+P_2\end{array}& \text{(s)}\frac{1}{2}[P_{n-1}+P_{n-2}]\end{array}$$
$\text{Which of the following is the only}\text{correct option?}$

1. $\left(c\right)\to \left(q\right)$
2. $\left(d\right)\to \left(r\right)$
3. $\left(a\right)\to \left(s\right)$
4. $\left(b\right)\to \left(r\right)$

Q. 16 people study French, 21 study Spanish and total number of students studying these languages is 30. Consider the experiment of radomnly selecting students and the events X and Y defined as
X = {Student studying French is selected}
Y = {Student studying Spanish is selected}
X and Y are mutually exclusive events.

1. True
2. False

Q. A player tosses a coin and scores one point for every head and two point for every tail that truns up. He plays on until his scores reaches or psses $n$. $P_n$ denotes the probability of getting a scores of exactly $n$
$$\begin{array}{cc}\text{List I}& \text{List II}\\ \begin{array}{cc}\text{(a)}& \text{the value of}{P}_n\text{is}\end{array}& \text{(p)}1\\ \begin{array}{c}\text{(b) the value of}{P}_n+\frac{1}{2}{P}_{n-1}\end{array}& \text{(q)}\frac{5}{4}\\ \begin{array}{c}\text{(c)}2P_{101}+P_{100}\end{array}& \text{(r)}2\\ \begin{array}{c}\text{(d)}{P}_1+P_2\end{array}& \text{(s)}\frac{1}{2}[P_{n-1}+P_{n-2}]\end{array}$$
$\text{Which of the following is the only}\text{incorrect option?}$

1. $\left(a\right)\to \left(s\right)$
2. $\left(c\right)\to \left(q\right)$
3. $\left(d\right)\to \left(q\right)$
4. $\left(b\right)\to \left(p\right)$

Q.
$A\mathrm{die}\mathrm{is}\mathrm{rolled}.\mathrm{Let}\mathrm{us}\mathrm{define}\mathrm{event}{E}_1\mathrm{as}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{possible}\mathrm{outcomes}\mathrm{where}\mathrm{the}\mathrm{number}\mathrm{on}\mathrm{the}\mathrm{face}\mathrm{of}\mathrm{the}\mathrm{die}\mathrm{is}\mathrm{even}\mathrm{and}\mathrm{event}{E}_2\mathrm{as}\mathrm{the}\mathrm{set}\mathrm{of}\mathrm{possible}\mathrm{outcomes}\mathrm{where}\mathrm{the}\mathrm{number}\mathrm{on}\mathrm{the}\mathrm{face}\mathrm{of}\mathrm{the}\mathrm{die}\mathrm{is}\mathrm{odd}.\mathrm{Event}{E}_1\mathrm{and}{E}_2\mathrm{are}\mathrm{mutually}\mathrm{exclusive}.$

1. True
2. False

Q.

A man firing at a distant target has 10% chance of hitting the target in one shot. The number of times he must fire at the target to have about 50% chance of hitting the target is

1. $10$

2. $9$

3. $7$

4. $8$

Q. Two dice are rolled. Let us define the following events
A: "sum of numbers on dice is greater than
or equal to 10".
B: "at least one number is 5".
Which of the following option(s) is/are correct?

1. Event A is a simple event.
2. Events A and B are not mutually exclusive events.
3. Event B is a compound event.
4. Events A and B are mutually exclusive events.

Q. Let ‘head’ means one and ‘tial’ means two and the coefficients of the equation $a{x}^2+bx+c=0$ are chosen by tossing a coin. The probability that the roots of the equation are non – real, is equal to:

1. $\frac{5}{8}$
2. $\frac{7}{8}$
3. $\frac{3}{8}$
4. $\frac{1}{8}$

Q. A four digit number (numbered from 0000 to 9999) is said to be lucky if sum of its first two digits is equal to the sum of its last two digits. If a four digit number is picked up at random, then the probability that it is a lucky number is:

1. 0.067
2. 0.67
3. None
4. 0.07

Q. Three athletes A, B and C participate in a race. Both A and B have the same probability of winning the race and each is twice as likely to win as C. The probability that B or C wins the race is

1. $\frac{2}{3}$
2. $\frac{3}{5}$
3. $\frac{3}{4}$
4. $\frac{13}{15}$

Q. If two different numbers are taken from the set $\{0,1,2,3,\cdots ,10\}$; then the probability that their sum as well as absolute difference are both multiple of 4, is:

1. $\frac{6}{55}$
2. $\frac{12}{55}$
3. $\frac{14}{45}$
4. $\frac{7}{55}$

Q. There is a group of 30 students from class XIth and class XIIth.
$EventX=\left\{\text{Students in class}X{I}^{th}\right\}$
$EventY=\left\{\text{Students in class}XI{I}^{th}\right\}$
X and Y are mutually exclusive events.

1. True
2. False

Q. $\begin{array}{cccc}& \text{List- I}& & \text{List-II}\\ \text{(I)}& {\log }_{x+1}(4x^3+9x^2+6x+1)+{\log }_{4x+1}(x^2+2x+1)& \text{(P)}& -1\\ & =5,\text{then the number of solution is}& & \\ \text{(II)}& \text{If}f\left(x\right)=\{\begin{array}{cc}{\left(\frac{3}{4}\right)}^{\frac{\tan 4x}{\tan 3x}},& 0<x<\frac{\pi }{2}\\ a+2,& x=\frac{\pi }{2}\\ (1+|\cot x|{)}^{ab|\tan x|},& \frac{\pi }{2}<x<\pi \end{array}& \text{(Q)}& 0\\ & \text{is continuous at}x=\frac{\pi }{2},\text{then}a+b=& & \\ \text{(III)}& \text{If}I=\underset{-1}{\overset{3}{\int }}\frac{1}{x^2}dx,\text{then}|3I|=& \text{(R)}& 1\\ \text{(IV)}& \text{If x-z plane divides the line joining of the points}& \text{(S)}& 2\\ & P(3,a,-4)\text{and}Q(2,-\frac{2}{3},5)\text{in the ratio}3:1,& & \\ & \text{then,}a=& & \\ & & \text{(T)}& 3\\ & & \text{(U)}& \text{Does not}\\ & & & \text{exist}\end{array}$

Which of the following is the only CORRECT combination?

1. $\left(I\right)\to \left(U\right)$
2. $\left(II\right)\to \left(P\right)$
3. $\left(IV\right)\to \left(Q\right)$
4. $\left(IV\right)\to \left(T\right)$

Q.

If A and B are two mutually exclusive and exhaustive events, then $P\left(B\right)=1-P\left(A\right).$

1. True

2. False

Q. If $\frac{1+3p}{3},\frac{1-p}{4}$ and $\frac{1-2p}{2}$ are probabilities of mutually exclusive events of a random experiment, then the range of $p$ is

1. $[\frac{1}{3},\frac{1}{2}]$
2. $[\frac{1}{4},\frac{1}{2}]$
3. $[\frac{1}{3},\frac{2}{3}]$
4. $[\frac{1}{3},\frac{2}{5}]$

Q. If m is a natural such that $m\le 5,$ then the probability that the quadratic equation $x^2+mx+\frac{1}{2}+\frac{m}{2}=0$ has real roots is

1. 1/5
2. 2/3
3. 3/5
4. 1

Q. Two events $A$ and $B$ have the probabilities $0.25$ and $0.5$ respectively. The probability that both $A$ and $B$ occur simultaneously is 0.14. Then the probability of neither A nor B occurs is equal to:

1. $0.61$
2. $0.39$
3. $0.29$
4. $0.19$