Axiomatic Approach

Q. In a precision bombing attack, there is $50\%$ chance that any one bomb will strike the target. Two precise hits are required to destroy the target completely. The number of bombs which should be dropped to give a $99\%$ chance or better of completely destroying the target can be

1. $12$
2. $11$
3. $10$
4. $13$

Q.

A man makes attempts to hit the target. The probability of hitting the target is $\frac{3}{5}$. Then the probability that $A$ hit the target exactly $2$ times in $5$ attempts is

1. $\frac{144}{625}$

2. $\frac{72}{3125}$

3. $\frac{216}{625}$

4. None of these

Q.

Out of $5$ apples, $10$ mangoes and $15$ oranges, any $15$ fruits are to be distributed among two persons. Then the total number of ways of distribution?

1. $66$

2. $36$

3. $60$

4. None of the above

Q. Two persons A and B agree to meet at a place between 11 am to 12 noon. The first one to arrive waits for 20 min and then leave. If the time of their arrival be independent and at random, then the probability that A and B meet is

1. $\frac{5}{9}$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $\frac{4}{9}$

Q.

Probability of solving specific problem independently by A and B arerespectively. If both try to solve the problem independently, find the probability that

(i) the problem is solved (ii) exactly one of them solves the problem.

Q.

$AandB$ are two independent events such that $P(AUB’)$ $=0.8$ and $P\left(A\right)=0.3$, then $P\left(B\right)$is

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

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

3. $\frac{3}{8}$

4. $\frac{1}{8}$

Q.

Each coefficient in the equation $a{x}^2+bx+c=0$ is determined by throwing an ordinary die. Find the probability that the equation will have equal roots.

1. $\frac{3}{216}$

2. $\frac{5}{216}$

3. $\frac{15}{216}$

4. $\frac{1}{216}$

Q.

If $A$ and $B$ are two independent events such that $P(A\cap B)=\frac{3}{25}$, and $P(A\cap B)=\frac{8}{25}$, then $P\left(A\right)=$

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

2. $\frac{3}{8}$

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

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

Q.

What is the difference between the Bayes theorem and conditional probability?

Q.

A box contains 100 bulbs, 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that :

(i) all 10 arc defective

(ii) all 10 are good

(iii) at least one is defective

(iv) None is defective

Q. A and B decide to meet at Hanuman temple between 5 pm and 6 pm with a condition that no one would wait for other more than 15 minutes. What is the probability that they meet?

Q.

A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that was produced by machine B?

Q. A fair coin is tossed a fixed number of times. If the probability of getting exactly $3$ heads equals the probability of getting exactly $5$ heads, then the probability of getting exactly one head is

1. $1/64$
2. $1/32$
3. $1/16$
4. $1/8$

Q. A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.

Q. A four digit number (numbered from 0000 to 9999) is said to be lucky if the sum of 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 lucky is

1. (a) >0.065
   

2. (b) <0.068
   

3. (c) 0.066
   

4. (d) 0.067

Q.

State which of the following are not the probability distributions of a random variable. Give reasons for your answer.

(i)

X	0	1	2
P (X)	0.4	0.4	0.2

(ii)

X	0	1	2	3	4
P (X)	0.1	0.5	0.2	− 0.1	0.3

(iii)

Y	−1	0	1
P (Y)	0.6	0.1	0.2

(iv)

Z	3	2	1	0	−1
P (Z)	0.3	0.2	0.4	0.1	0.05

Q. A fair coin is tossed $n-$times such that the probability of getting at least one head is at least $\mathrm{0.9.}$ Then the minimum value of $n$ is

Q.

A man make attempts to hit the target. The probability of hitting the target is 3/5. Then the probability that A hit the target exactly 2 times in 5 attempts is:

1. 

2. 

3. 

4. 

Q. A five digit number is formed by the digits 1, 2, 3, 4, 5, 6 and 8. The probability that the number has even digit at both ends is:

1. None
2. $\frac{3}{7}$
3. $\frac{2}{7}$
4. $\frac{4}{7}$

Q. If the letters of the word PROBABILITY are written down at random in a row, the probability that two B-s are together is

1. $\frac{2}{11}$
2. $\frac{10}{11}$
3. $\frac{3}{11}$
4. $\frac{6}{11}$

Q.

Find the probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two I's do not come together.

Q.

The letters of the word $ASSASSIN$ are written down at random in a row. The probability that no two $S$ occur together is

1. $\frac{1}{35}$

2. $\frac{1}{14}$

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

4. None of these

Q.

A person write 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is

1. $\frac{11}{24}$

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

3. $\frac{15}{24}$

4. $\frac{23}{24}$

Q. Two integers x and y are chosen with replacement out of the set {0, 1, 2, 3, . . . .10}. Then find the probability that |x - y|>5

1. 
2. 
3. 
4. 

Q.

A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is

(a) $\frac{3}{28}$
(b) $\frac{2}{21}$
(c) $\frac{1}{28}$
(d) $\frac{167}{168}$

Q. Let $E^C$ denote the complement of an event E. let E, F, G be pair wise independent events with P(G) > 0 and $P(E\cap F\cap G)=0$. Then $P\left(\frac{E^C\cap F^C)}{G}\right)$ equals

1. 
2. 
3. 
4. 

Q. Two numbers $a$ and $b$ are chosen at random from the first $30$ natural numbers. The probability that $a^2-b^2$ is divisible by $3$ is

1. $\frac{47}{87}$
2. $\frac{37}{87}$
3. $\frac{57}{87}$
4. $\frac{33}{87}$

Q.

If $P(A\cup B)=P(A\cap B)$ for any two events A and B. then

1. None of these

2. $P\left(A\right)=P\left(B\right)$

3. $P\left(A\right)>P\left(B\right)$

4. $P\left(A\right)<P\left(B\right)$

Q.

A class consists of 80 students, 25 of them are girls 55boys 10of them are rich and remaining poor 20 of them are fair complexioned. The probability of selecting a fair complexioned rich girl is

Q.

Let $x$ be a set containing $n$elements. If two subsets$A$ and $B$of $x$picked at random, the probability that$A$ and $B$have the same number of elements, is

1. $\frac{{}^{2n}{\mathrm{ℂ}}_n}{2^{2n}}$

2. $\frac{1}{{}^{2n}{\mathrm{ℂ}}_n}$

3. $\frac{\left[1.3.5...\left(2n-1\right)\right]}{2^n}$

4. $\frac{3^n}{4^n}$