Classical probability

Let S be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from S, is a multiple of 7 but not divisible by 5, then $9p$ is equal to

A six faced die is biased such that

$3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1)$.

Let X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of X is :

Let S = {1, 2, 3, ..., 2022}. Then the probability, that a randomly chosen number n from the set S such that HCF (n, 2022) = 1, is :

The value of $\dfrac{625}{4}p_1$ is ____.

The value of $\dfrac{125}{4}p_2$ is ____.

Let $E$, $F$ and $G$ be three events having probabilities
$$P(E) = \frac18,\quad P(F) = \frac16\quad\text{and}\quad P(G) = \frac14,\quad\text{and let } P(E\cap F\cap G) = \frac1{10}.$$
For any event $H$, if $H^c$ denotes its complement, then which of the following statements is (are) TRUE?

A number is chosen at random from the set $\{1, 2, 3, \ldots, 2000\}$. Let $p$ be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of $500p$ is ____.

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least three successful hits are required. Then the minimum number of missiles that have to be fired so that the probability of completely destroying the target is NOT less than 0.95, is ______

Two fair dice, each with faces numbered 1, 2, 3, 4, 5 and 6, are rolled together and the sum of the numbers on the faces is observed. This process is repeated till the sum is either a prime number or a perfect square. Suppose the sum turns out to be a perfect square before it turns out to be a prime number. If $p$ is the probability that this perfect square is an odd number, then the value of $14p$ is ______

The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and NONE of the remaining students gets the seat previously allotted to him/her is

For $i = 1, 2, 3, 4$, let $T_i$ denote the event that the students $S_i$ and $S_{i+1}$ do NOT sit adjacent to each other on the day of the examination. Then, the probability of the event $T_1 \cap T_2 \cap T_3 \cap T_4$ is

Three randomly chosen nonnegative integers $x$, $y$ and $z$ are found to satisfy the equation $x + y + z = 10$. Then the probability that $z$ is even, is

For three events A, B and C,

P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = $\dfrac14$ and P(All the three events occur simultaneously) = $\dfrac1{16}$.

Then the probability that at least one of the events occurs, is :

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 multiple of 4, is :

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :

From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for these committees to have more women than men, is :

$P(X>Y)$ is

$P(X=Y)$ is

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is

A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\dfrac13$, then the correct option(s) with the possible values of $n_1$ and $n_2$ is(are)