Complement of an Event

Q. If each of the letters in the English alphabet is assigned odd numerical value beginning $A=1,$ $B=3$ and so on, then the total numerical value of the letters of the word $INDIAN$ is

1. $86$
2. $88$
3. $89$
4. $96$

Q.

If $A$ and $B$ are any two events, then the probability that exactly one of them occur is

1. $P\left(A\right)+P\left(B\right)-P(A\cap B)$

2. $P\left(A\right)+P\left(B\right)-2P(A\cap B)$

3. $P\left(A\right)+P\left(B\right)-P\left(AUB\right)$

4. $P\left(A\right)+P\left(B\right)-2P\left(AUB\right)$

Q.

Let $A$and $B$ be two events such that $P\left(A\right)=0.3$ and $P(A\cup B)=0.8$. If $A$and $B$ are independent events, then $P\left(B\right)$ is

1. $\frac{5}{6}$

2. $\frac{5}{7}$

3. $\frac{3}{5}$

4. $\frac{2}{5}$

Q.

If $f\left(x\right)=\frac{k}{2^x}$ is a probability distribution of a random variable $X$ that can take on the values $x=0,1,2,3,4$. Then $k$ is equal to

1. $\frac{16}{15}$

2. $\frac{15}{16}$

3. $\frac{31}{16}$

4. None of these

Q. If A and B are independent events such that P(A) = p, P(B) = 2p and P(Exactly one of A and B occurs) = $\frac{5}{9}$, then find the value of p.

Q.

If A and B are independent events, such that P(A U B) = 0.7, P(B) = 0.5, then P(A) = ……

1. 0.2

2. 0.6

3. 0.4

4. 0.3

Q.

The probability that a certain kind of component will survive a given shock test is $\frac{3}{4}$ The probability that exactly $2$ of the next $4$ components tested survive is

1. $\frac{9}{41}$

2. $\frac{25}{128}$

3. $\frac{1}{5}$

4. $\frac{27}{128}$

Q. Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is :

1. $\frac{1}{8}$
2. $1$
3. $\frac{5}{8}$
4. $\frac{5}{16}$

Q. Let $9$ distinct balls be distributed among $4$ boxes, $B_1,B_2,B_3$ and $B_4$. If the probability that $B_3$ contains exactly $3$ balls is $k{\left(\frac{3}{4}\right)}^9$, then $k$ lies in the set

1. $\{x\in R:|x-3|<1\}$
2. $\{x\in R:|x-1|<1\}$
3. $\{x\in R:|x-5|\le 1\}$
4. $\{x\in R:|x-2|\le 1\}$

Q. If a fair coin is tossed $5$ times, then the probability that we get exactly $2$ heads is
$\left(a\right)\frac{5}{16}\left(b\right)\frac{5}{32}$
$\left(c\right)\frac{3}{8}\left(d\right)\frac{1}{4}$

Q. If A and B are two events such that , find P (not A and not B).

Q.

If P (A)= $\frac{3}{5}$ and P (B)= $\frac{1}{5}$, find P $A\cap B$, if A and B are independents events.

Q.

The probability of at least one double-six being thrown in n throws with two ordinary dice is greater than 99%. Calculate the least numerical value of n.___

Q.

A factory produces bulbs. The probability that any one bulb is defective is $\frac{1}{50}$ and they are packed in 10 boxes. From a single box, find the probability that

(i) None of the bulbs is defective.

(ii) Exactly two bulbs are defective.

Q.

Consider two events A and B such that $P\left(A\right)=\frac{1}{4}$, $P\left(\frac{B}{A}\right)=\frac{1}{2}$, $P\left(\frac{A}{B}\right)=\frac{1}{4}$. For each of the following statements, which is true.

$\mathrm{I}.\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)=\frac{3}{4}$

$\mathrm{II}.$ The events A and B are mutually exclusive

$\mathrm{III}.\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)+\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)=1$

1. $\mathrm{I}\mathrm{only}$

2. $\mathrm{I}\&\mathrm{II}$

3. $\mathrm{I}\&\hspace{0.17em}\mathrm{III}$

4. $\mathrm{II}\&\mathrm{III}$

Q.

Three mangoes and three apples in a box. If two fruits are chosen at random, the probability that one is a mango and the other is an apple is

1. $\frac{2}{3}$

2. $\frac{3}{5}$

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

4. None of these

Q.

Random variable is a real valued function whose domain is the sample space of a ……… and range is the set of ……...

1. Real numbers, real numbers

2. Random variable, non-negative integers

3. Random experiment, real numbers

4. Random variable, real numbers

Q.

Let $P$ denotes the probability that a man aged $x$ years will die in a year. The probability that out of $n$ men $A_1,A_2,A_3,...,A_n$ each aged$X,A_1$ will die in a year and will be the first to die is

1. $\frac{1}{n[1-{(1-p)}^n]}$

2. $[1-{(1-p)}^n]$

3. $\frac{1}{(n-1)}[1-{(1-p)}^n]$

4. None of these

Q. If $P\left(A\right)=\frac{4}{5}$ and $P(A\cap B)=\frac{7}{10}$, then the value of $P\left(B|A\right)$ is

$\left(a\right)\frac{7}{10}\left(b\right)\frac{4}{5}\left(c\right)\frac{7}{8}\left(d\right)\frac{4}{7}$

Q.

Compute P$\left(\frac{A}{B}\right)ifP\left(B\right)=0.5P(A\cap B)=0.32$

Q.

The probability of rain today is $60$ percent$,$ and the independent probability of rain tomorrow is $75$ percent$.$ Calculate the probability that it will rain neither today nor tomorrow$.$

1. $10\%$

2. $15\%$

3. $45\%$

4. $55\%$

Q. The probability that an event $A$ occurs in a single trial of an experiment is $0.6$. In the first three independent trials of the experiment, the probability that $A$ occurs at least once is

1. $0.930$
2. $0.936$
3. $0.925$
4. $0.927$

Q. All the face cards are removed from a well-shuffled pack of $52$ cards. Out of the remaining cards, $4$ cards are drawn at random. The probability that they belong to different suits is

1. $\frac{{}^{13}{C}_4}{{}^{40}{C}_4}$
2. $\frac{{10}^4}{{}^{52}{C}_4}$
3. $\frac{{13}^4}{{}^{40}{C}_4}$
4. $\frac{{10}^4}{{}^{40}{C}_4}$

Q. If $P\left(A\right)=\frac{2}{5}$, $P\left(B\right)=\frac{3}{10}$ and $P(A\cap B)=\frac{1}{5}$, then find the value of $P\left(A^{\prime }|B^{\prime }\right).P\left(B^{\prime }|A^{\prime }\right)$.

Q. A box contains $12$ mangoes out of which $5$ are rotten and rest are good. Two mangoes are randomly taken out together. If it is known that atleast one of them is good, then the probability that both are good is $\frac{p}{q}$, where $p,q$ are coprime numbers. Then the value of $|p-q|$ is

Q.

Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both the cards are black.

Q.

It is known that 10% of certain articles manufactured are defective, what is the probability that in a random sample of 12 such articles, 9 are defective?

Q.

If $P\left(A\right)=\frac{1}{12},P\left(B\right)=\frac{5}{12}$and $P\left(B/A\right)=\frac{1}{15}$, then $P(A\cup B)$is equal to

1. $\frac{89}{180}$

2. $\frac{90}{180}$

3. $\frac{91}{180}$

4. $\frac{92}{180}$

Q. In a certain town , $60$% of the families own a car, $30$% own a house and $20$% own both a car and a house. If a family is randomly chosen, what is the probability that this family owns a car or a house but not both?

1. $0.7$
2. $0.9$
3. $0.5$
4. $0.1$

Q.

The probability that a student will succeed in the IIT entrance test is $0.2$ and that he will succeed in the Roorkee entrance test is $0.5$. If the probability that he will be successful at both the places is $0.3$, then the probability that he does not succeed at both places is

1. $0.4$

2. $0.3$

3. $0.2$

4. $0.6$