Multiplication Rule

Q. In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to:

1. $\frac{150}{6^5}$
2. $\frac{175}{6^5}$
3. $\frac{200}{6^5}$
4. $\frac{225}{6^5}$

Q. In an objective paper, there are two sections of $10$ questions each. For "section $1$", each question has $5$ options and only one option is correct and "section $2$" has $4$ options with multiple answers and marks for a question in this section is awarded only if he ticks all correct answers. Marks for each question in "section $1$" is $1$ and in "section $2$" is $3.$
(There is no negative marking)
If a canidate in total attempts $4$ questions all by guessing, then the probability of scoring $10$ marks is

1. $\frac{4}{15}\times (\frac{1}{15}{)}^3$
2. $\frac{1}{15}\times (\frac{14}{15}{)}^3$
3. $\frac{1}{15}\times (\frac{1}{15}{)}^3$
4. None of these

Q. $A$ can choose a lottery ticket out of three in which there is only $1$ prize and $2$ blanks and $B$ can choose $3$ lottery tickets out of $9$ in which there are $3$ prizes and $6$ blanks. If winning at least one prize is a success, then the ratio of probability of $A$'s success to that of $B$'s success is

1. $2:3$
2. $3:8$
3. $9:19$
4. $7:16$

Q. Two numbers are selected randomly from the set S = {1, 2, 3, 4, 5, 6} without replacement one by one. The probability that minimum of the two numbers is less than 4, is

1. $\frac{1}{5}$
2. $\frac{4}{5}$
3. $\frac{1}{15}$
4. $\frac{14}{15}$

Q.

Box I contain three cards bearing numbers 1, 2, 3; box II contains five cards bearing numbers 1, 2, 3, 4, 5; and box III contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box $i=1,2,3.$

The probability that $x_1,x_2$ and $x_3$ are in arithmetic progression, is ?

1. $\frac{9}{105}$

2. $\frac{10}{105}$

3. $\frac{11}{105}$

4. $\frac{7}{105}$

Q. $A$ can choose a lottery ticket out of three in which there is only $1$ prize and $2$ blanks and $B$ can choose $3$ lottery tickets out of $9$ in which there are $3$ prizes and $6$ blanks. If winning at least one prize is a success, then the ratio of probability of $A$'s success to that of $B$'s success is

1. $2:3$
2. $3:8$
3. $9:19$
4. $7:16$

Q. A pair of fair dice is rolled together till a sum of either $5$ or $7$ is obtained. Then the probability that $5$ before $7$ is

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. 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. In an objective paper, there are two sections of $10$ questions each. For "section $1$", each question has $5$ options and only one option is correct and "section $2$" has $4$ options with multiple answers and marks for a question in this section is awarded only if he ticks all correct answers. Marks for each question in "section $1$" is $1$ and in "section $2$" is $3.$
(There is no negative marking)
If a canidate in total attempts $4$ questions all by guessing, then the probability of scoring $10$ marks is

1. $\frac{1}{15}\times (\frac{1}{15}{)}^3$
2. $\frac{4}{15}\times (\frac{1}{15}{)}^3$
3. $\frac{1}{15}\times (\frac{14}{15}{)}^3$
4. None of these