Complement of a Set

Q.

The set $(A\cup B^{\prime }{)}^{\prime }\cup (B\cap C)$ is equal to

1. $A^{\prime }\cup B\cap C$

2. $A^{\prime }\cup B$

3. $A^{\prime }\cup C$

4. $A^{\prime }\cap B$

Q.

Three events A, B and C have probabilities $\frac{2}{5},\frac{1}{3}$ and $\frac{1}{2}$, respectively. If,

$P(A\cap C)=\frac{1}{5}andP(B\cap C)=\frac{1}{4}$, then find the values of $P\left(\frac{C}{B}\right)andP(A^{\prime }\cap C^{\prime }).$

Q. Let $U$ be universal set of all the students of class $XI$ of a co-educational school and $A$ be the set of all girls in class $XI$. Find $A^{\prime }$.

Q. Given two events $A$ and $B.$ If odds against $A$ are as $2:1$ and those in favour of $A\cup B$ are as $3:1,$ then

1. $\frac{1}{6}\le P\left(B\right)\le \frac{3}{4}$
2. $\frac{5}{12}\le P\left(B\right)\le \frac{3}{4}$
3. $\frac{1}{4}\le P\left(B\right)\le \frac{3}{5}$
4. $\frac{1}{8}\le P\left(B\right)\le \frac{3}{4}$

Q. If $E_1,E_2$ are two events with $E_1\cap E_2=\varphi ,$ then $P({\overline{E}}_1\cap {\overline{E}}_2)=$

1. $P\left({\overline{E}}_1\right)-P\left(E_2\right)$
2. $P\left({\overline{E}}_2\right)-P\left(E_1\right)$
3. $P\left(E_2\right)-P\left(E_1\right)$
4. $P\left({\overline{E}}_2\right)-P\left({\overline{E}}_1\right)$

Q. Let $A=\{2,4,6,8\}$ and $B=\{6,8,10,12\}$. Find $A\cup B$

Q. Let A and B be two events such that $P\left(A\right)=\frac{1}{3},P\left(\frac{A}{B}\right)=\frac{1}{2}$ and $P\left(\frac{B}{A}\right)=\frac{2}{5}$

$P(A\cup B)=$

1. $\frac{1}{3}$
2. $\frac{7}{15}$
3. $\frac{2}{15}$
4. $\frac{3}{15}$

Q. Given that $E$ and $F$ are events such that $P\left(E\right)=0.6,P\left(F\right)=0.3$ and $P(E\cap F)=0.2$, find $6P\left(F|E\right)$.

Q. If $A,B,C$ are mutually independent events, such that $0<P\left(A\right),P\left(B\right),P\left(C\right)<1,$ then which of the following statement(s) is/are correct ?
$I.A$ and $B\cup C$ are independent events.
$II.A$ and $B\cap C$ are independent events.

1. Only $II$ is true
2. Both are true
3. Both are false
4. Only $I$ is true

Q.

If A and B are two events such that

$P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$ and $P(A\cap B)=\frac{1}{4},$ then find

(iii) $P\left(\frac{A^{\prime }}{B}\right).$

Q. Let $A,B$ & $C$ be $3$ arbitary events defined on a sample space $S$ and if, $P\left(A\right)+P\left(B\right)+P\left(C\right)=p_1,P(A\cap B)+P(B\cap C)+P(C\cap A)=p_2$ & $P(A\cap B\cap C)=p_3$, then the probability that exactly one of the three events occurs is given by:

1. $p_1-p_2+p_3$
2. $p_1-p_2+2p_3$
3. $p_1-2p_2+p_3$
4. $p_1-2p_2+3p_3$

Q.

If A and B are two events such that

$P\left(A\right)=\frac{1}{2},P\left(B\right)=\frac{1}{3}$ and $P(A\cap B)=\frac{1}{4},$ then find

$P\left(\frac{A^{\prime }}{B^{\prime }}\right)$

Q. Let $n\left(U\right)=700$, $n\left(A\right)=200$, $n\left(B\right)=300$ and $n(A\cap B)=100,\text{then}n(A^c\cap B^c)$ is -

1. $400$
2. $300$
3. $600$
4. $200$

Q. A die is tossed thrice. Find the probability of getting an odd number at least once.

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below:

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}{E}_1\text{and}{E}_2\text{be mutually exclusive events, then}& 1& E_1\cap E_2=E_1\\ \text{(B)}& \text{If}{E}_1\text{and}{E}_2\text{are mutually exclusive and exhaustive events, then}& 2& (E_1-E_2)\cup (E_1\cap E_2)=E_1\\ \text{(C)}& \text{If}{E}_1\text{and}{E}_2\text{have common outcomes, then}& 3& E_1\cap E_2=\varphi ,E_1\cup E_2=S\\ \text{(D)}& \text{If}{E}_1\text{and}{E}_2\text{are events such that}{E}_1\subset E_2\text{, then}& 4& E_1\cap E_2=\varphi \end{array}$

Which of the following is the only CORRECT combination?

1. $A\to 1,B\to 3,C\to 2,D\to 4$
2. $A\to 4,B\to 3,C\to 2,D\to 1$
3. $A\to 4,B\to 2,C\to 3,D\to 1$
4. $A\to 3,B\to 4,C\to 1,D\to 2$

Q. If $P\left(A\right)=\frac{6}{11},P\left(B\right)=\frac{5}{11}$ and $P(A\cup B)=\frac{7}{11}$, find
(i) $P(A\cap B)$
(ii) $P\left(A|B\right)$
(iii) $P\left(B|A\right)$

1. $0.80,0.98,0.87$
2. $0.58,0.58,0.83$
3. $0.42,0.67,0.57$
4. $0.36,0.80,0.66$

Q. If $P\left(A\right)=0.8,P\left(B\right)=0.5$ and $P\left(B|A\right)=0.4$, find
(i) $P(A\cap B)$
(ii) $P\left(A|B\right)$
(iii) $P(A\cup B)$

Q. Find the interval in which the function $f\left(x\right)=10-6x-2x^2$ is strictly increasing or decreasing

Q. If $U=\{1,2,3,4,5,6,7,8,9,10\},A=\{1,2,3,5\},$

$B=\{2,4,6,7\}$ and $C=\{2,3,4,8\}$ . Then

(ii) $(C-A{)}^{\prime }$ = ____

Q.

If A, B and C are three events such that P(B) =$\frac{3}{4},P(A\cap B\cap C^{\prime })=\frac{1}{3}andP(A^{\prime }\cap B\cap C^{\prime })=\frac{1}{3},$ then $P(B\cap C)$ is equal to

1. $\frac{1}{12}$

2. $\frac{1}{6}$

3. $\frac{1}{15}$

4. $\frac{1}{9}$

Q. Prove that number of subsets of a set containing $n$ distinct elements is $2^n$, for all $n\in N$.

Q.

Q. Let $A$ and $B$ be the events such that $P\left(A\right)=\frac{7}{13},P\left(B\right)=\frac{9}{13}$ and $P(A\cap B)=\frac{4}{13}$
Find $P\left(\overline{B}/\overline{A}\right)$