Total Probability Theorem

Q.

The students $S_1,S_2,..........S_{10}$ are to be divided into $3$ groups $A,B$ and $C$ such that each group has at least one student and the group $C$ has at most $3$ students. Then the total number of possibilities of forming such groups is

Q.

From a book containing 100 pages, one page is selected randomly. The probability that the sum of the digits of the page number of the selected page is 11, is

1. 

2. None of these

3. 

4. 

Q.

What are the two laws of probability?

Q. When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}.$ The probability that the missile hits the target, given that it is not intercepted is $\frac{3}{4}.$ If three missiles are fired independently from the ship, then the probability that all three hit the target, is

1. $\frac{1}{12}$
2. $\frac{1}{8}$
3. $\frac{3}{8}$
4. $\frac{3}{4}$

Q.

From a group of $7$ men and $4$ ladies a committee of $6$ persons is formed, then the probability that the committed contains $2$ ladies is:

1. $\frac{5}{13}$

2. $\frac{5}{11}$

3. $\frac{4}{11}$

4. $\frac{3}{11}$

Q.

A coin is tossed twice. If events $A$ and $B$ are defined as $A$ = head on first toss, $B$ = head on second toss. Then the probability of $A\cup B=$

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

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

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

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

Q.

From a pack of $52$ cards, two are drawn with replacement.

The probability, that the first is a diamond and the second is a king is:

1. $\frac{1}{26}$

2. $\frac{17}{2704}$

3. $\frac{1}{52}$

4. None of these.

Q.

An urn contains m white and n black balls. A ball is drawn at randow and is put back into the urn along with K additional balls of the same colour as that of the ball drawn. A ball is again drawn at randow. Show that the probability of drawing a white ball now does not depend on k.

Q. Each of three identical jewellery boxes has two drawers. In each drawer of the first box, there is a gold watch. In each drawer of the second box, there is a silver watch. In one drawer of the third box, there is a gold watch while in the other, there is a silver watch. If we select a box at random, open one of the drawers and find it to contain a silver watch, then the probability that the other drawer has the gold watch in it, is

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

Q. Three dice thrown together. Find the probability of getting a total of at least 6?

Q. If two loaded dice each have the property that $2$ and $4$ is three times as likely to appears as $1,3,5\text{or}6$ on each roll. When two such dice are rolled, the probability of obtaining a total of $7$ is $p$, then the value of $[\frac{1}{p}]$ is (where $\left[x\right]$ represents the greates integer less than or equal to $x$.)

Q.

A coin is tossed m + n times, where $m\ge n.$ The probability of getting at least m consecutive heads is

1. $\frac{n+1}{2^{m+1}}$
2. $\frac{n+2}{2^{m+1}}$
3. None of these
4. $\frac{m+2}{2^{n+1}}$

Q. If $A$ and $B$ are two events of a sample space $S$ such that $P\left(A\right)=0.2,P\left(B\right)=0.6$ and $P\left(A|B\right)=0.5$ then $P\left(A^{\prime }|B\right)=$

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

Q. The probability that the birthday of six different persons will fall in exactly two calendar months is

1. $\frac{134}{{12}^5}$
2. $\frac{143}{{12}^5}$
3. $\frac{341}{{12}^5}$
4. $\frac{331}{{12}^5}$

Q.

6 married couples are present in a room. If 4 people are chosen at random, then the chance that exactly one married couple is among the 4 is?

1. 16/66

2. 16/33

3. 23/33

4. 8/33

Q. A die is loaded so that the probability of a face $i$ is proportional to $i,i=1,2,...6$. The probability of an even number occurring when the die is rolled is $\frac{b}{7},$ the value of $b$ is

Q.

There are $10$ pairs of shoes in a cupboard from which $4$ shoes are packed at random. The probability that there is at least one pair is

1. $\frac{99}{323}$

2. $\frac{224}{323}$

3. $\frac{100}{323}$

4. None of these

Q. The probability that in a family of $5$ members, exactly two members have birthday on Sunday is

1. $\frac{(12\times 5^3)}{7^5}$
2. $\frac{(10\times 6^2)}{7^5}$
3. $\frac{(10\times 5^3)}{7^5}$
4. $\frac{(10\times 6^3)}{7^5}$

Q. There are two bags, one of which contains 5 black and 4 white balls while the other contains 3 black and 6 white balls. A die is thrown. If it shows up 1 or 3, a ball is taken from the first bag. But if it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a white ball.

1. $\frac{1}{2}$
2. $\frac{14}{27}$
3. $\frac{4}{27}$
4. $\frac{16}{27}$

Q. Five different games are to be distributed among 4 children randomly. The probability that each child get at least one game is

1. 
2. 
3. 
4. 

Q.

A box contains $6$ nails and $10$ nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or is a nail

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

2. $\frac{5}{16}$

3. $\frac{11}{16}$

4. $\frac{14}{16}$

Q. A bag contains $(2n+1)$ coins. It is known that $n$ of these coins have a head on both sides, whereas the remaining $(n+1)$ coins are fair. A coin is selected at random from the bag and tossed once. If the probability that the toss results in a head is $\frac{31}{42},$ then $n$ is equal to

1. $12$
2. $10$
3. $7$
4. $8$

Q. If $5$ distinct balls are placed at random into $5$ cells, then the probability that exactly one cell remains empty is

1. $\frac{8}{125}$
2. $\frac{1}{125}$
3. $\frac{48}{125}$
4. $\frac{12}{125}$

Q.

Find the probability of throwing at most 2 sixes in 6 throws of a single die.

Q.

A company manufacturers video games with a current defect rate of $0.95\%$. To make sure as few defective video games are delivered as possible, they are tested before delivery. The test is $98\%$ accurate at determining if a video game is defective. If $100000$ products are manufactured and delivered in a month, approximately how many defective products are expected to be delivered?

Q. If $10$ objects are distributed at random among $10$ persons, then the probability that at least one of them will not get anything is

1. $\frac{10!}{{10}^{10}}$
2. $\frac{{10}^{10}-10!}{{10}^{10}}$
3. $\frac{1}{{10}^{10}}$
4. $\frac{1}{10}$

Q. The probability of getting 11 when an ordinary die is thrown twice is......

Q. An artillery target may be either at point I with probability $\frac{8}{9}$ or at point II with probability $\frac{1}{9}$. We have 55 shells, each of which can be fired either at point I or II. Each shell may hit the target, independent of the other shells, with probability $\frac{1}{2}$. Maximum number of shells that must be fired at point I to have maximum probability is

1. $20$
2. $25$
3. $29$
4. $35$

Q. Twelve balls are distributed among three boxes. The probability that the first box will contains three balls.

1. $\frac{2^9}{3^{12}}$
2. $\frac{{}^{12}{C}_3}{3^{12}}$
3. $\frac{{}^{12}{C}_3\times 2^9}{3^{12}}$
4. $\frac{{}^{12}{C}_3\times 2^9}{{12}^3}$

Q. Four distinct integers are picked at random from $\{0,1,2,3,4,5,6\}.$ If the probability that among those selected, the second smallest is $3,$ is $p,$ then $p$ is equal to

1. $\frac{9}{35}$
2. $\frac{1}{35}$
3. $\frac{3}{35}$
4. $\frac{6}{35}$