Multiplication Rule

Q.

A bag contains $4$white and $3$red balls. Two draws of one ball each are made without replacement. That the probability that both the balls are red is

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

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

3. $\frac{3}{7}$

4. $\frac{4}{7}$

Q. Four persons $A,B,C$ and $D$ throw an unbiased die, turn by turn, in succession till one gets an even numbers and win the game. What is the probability that $A$ wins if $A$ begins?

1. $\frac{8}{15}$
2. $\frac{1}{4}$
3. $\frac{1}{2}$
4. $\frac{7}{12}$

Q.

If two different numbers are taken from the set $0,1,2,3,......,10;$ then the probability that their sum, as well as absolute difference, are both multiples of$4$, is.

1. $\frac{12}{55}$

2. $\frac{14}{45}$

3. $\frac{7}{55}$

4. $\frac{6}{55}$

Q. If three students A, B, C independently solve a problem with probabilities $\frac{1}{3},\frac{1}{4}$ and $\frac{1}{5}$ respectively, then the probability that the problem will be solved is

1. $\frac{3}{5}$
2. $\frac{4}{5}$
3. $\frac{47}{60}$
4. $\frac{2}{5}$

Q.

An experiment succeeds twice as often as it fails. find the probability that in $4$ trails there will be at least three success

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

2. $\frac{8}{27}$

3. $\frac{16}{27}$

4. $\frac{24}{27}$

Q. A student appeared in an examination consisting of $8$ true - false type questions. The student guesses the answers with equal probability. The smallest value of $n$, so that the probability of guessing at least $n$ correct answers is less than $\frac{1}{2}$, is

1. $4$
2. $3$
3. $5$
4. $6$

Q.

A signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively, is received by station $A$ and then transmitted to station $B$. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station $B$ is green, then the probability that the original signal green is

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

2. $\frac{6}{7}$

3. $\frac{20}{23}$

4. $\frac{9}{20}$

Q. Find the probability of getting 9 cards of the same suit in one hand at a game of bridge?

Q.

If a party of $\text{n}$ persons sit at a round table, then the odds against two specified individuals sitting next to each other are

1. $\text{2:(n-3)}$

2. $\text{(n-3):2}$

3. $\text{(n-2):2}$

4. $\text{2:(n-2)}$

Q. A flashlight has 10 batteries out of which 4 are dead. If 3 batteries are selected without replacement and tested, then the probability that all 3 are dead is

1. $\frac{1}{30}$
2. $\frac{1}{15}$
3. $\frac{1}{10}$
4. $\frac{2}{8}$

Q.

A box contains $24$ identical balls, of which $12$ are white and $12$ are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the $4th$ time on the $7th$ draw is

1. $\frac{5}{64}$

2. $\frac{27}{32}$

3. $\frac{5}{32}$

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

Q. There are $3$ bags A, B and C. Bag A contains $1$ Red and $2$ Green balls, bag B contains $2$ Red and $1$ Green balls and bag C contains only one Green ball. One ball is drawn from bag A and put into bag B then one ball is drawn from bag B and put into bag C and finally one ball is drawn from bag C and put into bag A. When this operation is completed, probability that bag A contains $2$ Red and $1$ Green balls, is -

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

Q. A man speaks truth $2$ out of $3$ times. He picks one of the natural numbers in the set $S=\{1,2,3,4,5,6,7\}$ and reports that it is even. The probability that it is actually even is

1. $\frac{1}{10}$
2. $\frac{2}{5}$
3. $\frac{3}{5}$
4. $\frac{1}{5}$

Q. A five-digit number is written down at random. The probability that the number is divisible by $5$ and no two consecutive digits are identical, is:

1. $\frac{1}{5}$
2. $\frac{34}{90}.{\left(\frac{9}{10}\right)}^3$
3. ${\left(\frac{3}{5}\right)}^4$
4. ${\left(\frac{2}{5}\right)}^4$

Q. A bag contains $15$ red and $20$ black balls. Each ball is numbered $1,2\text{or}3$. $20\%$ of red balls are numbered $1$ and $40\%$ are numbered $3$. Similarly $45\%$ of black balls are numbered $2$ and $30\%$ are numbered $3$. One balls is drawn at random and found to be numbered $2$, then probability that it was red ball is

Q.

100 students appeared for two examination. 60 passed the first, 50 passed

the second and 30 passed both. Find the probability that a student

selected at random has passed at least one examination.

Q. Whenever horses $a,b,c$ race together, their respective probabilities of winning the race are $0.3,0.5$ and $0.2$ respectively. If they race three times the probability that "the same horse wins all the three races" is $p$ and the probability that $a,b,c$ each wins one race is $q$, then the value of $\frac{p}{q}$ is (Assume no dead heat)

1. $\frac{9}{8}$
2. $\frac{8}{9}$
3. $\frac{3}{2}$
4. $\frac{2}{3}$

Q. A set of three numbers are chosen from the set $S=\{1,2,3,\dots ,(2n+1\left)\right\}.$ If the probability that the numbers chosen are in arithmetic progression is $\frac{4}{21},$ then the value of $n$ is

Q. From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that both of them are kings.

Q.

If the probability of $X$ to fail in the examination is$0.3$ and that for $Y$ is $0.2$, then the probability that either $X$ or $Y$ fail in the examination is

1. $0.5$

2. $0.44$

3. $0.6$

4. $noneofthese$

Q. Three cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that the cards are a king, a queen and a jack.

Q. The probability of a man hitting a target is $\frac{1}{10}$. The least number of shots required, so that the probability of his hitting the target at least once is greater than $\frac{1}{4}$, is

Q. In a bag, there are 6 balls of which 3 are white and 3 are black. 6 balls are drawn successively $\left(i\right)$ without replacement, $\left(ii\right)$ with replacement. What is the chance that the colors are alternate?

1. $\frac{1}{20}$; $\frac{1}{64}$
2. $\frac{1}{10}$; $\frac{1}{64}$
3. $\frac{1}{20}$; $\frac{1}{32}$
4. $\frac{1}{10}$; $\frac{1}{32}$

Q.

A man draws a card from a pack of 52 playing cards, replaces it and shuffles

the pack. He continues this processes until he gets a card of spade. The

probability that he will fail the first two times is [MNR 1980]

1. 

2. 

3. 

4. None of these

Q.

At least number of times a fair coin must be tossed so that the probability of getting at least one head is at least 0.8, is

1. 7

2. 6

3. 5

4. None of these

Q.

If the probability for A to fail in an examination is 0.2 and that for B is 0.3, then the probability that either A or B fails is

1. > 0.5

2. 0.5

3. 0

4. $\ge$ 0.5

Q. Amit and Nisha appear for an interview for two vacancies in a company. The probability of Amit's selection is $\frac{1}{5}$ and that of Nisha's selection is $\frac{1}{6}$. Then the probability that both of them are selected is:

1. $\frac{1}{20}$
2. $\frac{1}{25}$
3. $\frac{1}{30}$
4. $\frac{1}{10}$

Q. A game is played with fair cubic die which has one red side, two blue sides and three green sides. The result is the colour of the top side after the die is rolled. If the die is rolled repeatedly, the probability that second blue result occurs on or before the tenth roll can be expressed in the form of $\frac{3^p-2^q}{3^r}$ where $p,q,r$ are positive integers. Then the value of $p+r-q$ is

Q. If $12$ identical coins are distributed among three children at random. The probability of distributing so that each child gets atleast two coins is $\frac{7k}{3^{12}}$ then $k$ is

Q. A slip of paper is given to $A$, who marks it with either a plus or a minus sign; the probability of his writing a plus is $\frac{1}{3}$. He then passes the slip to $B$, who may either leave it or change the sign before passing it on to $C$. Next $C$ passes the slip to $D$ after perhaps changing the sign; finally $D$ passes it to an honest judge after perhaps changing the sign. The judge sees a plus sign on the slip. It is known that $B,C$ and $D$ each change the sign with probability $\frac{2}{3}$. Then probability that $A$ originally wrote a plus is $\frac{a}{b}$ (where $a,b$ are coprime numbers), then the value of $a+b$ is