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 :

Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is :

Let A and B be two events such that $P(B|A)=\dfrac25$, $P(A|B)=\dfrac17$ and $P(A\cap B)=\dfrac19$. Consider

(S1) $P(A'\cup B)=\dfrac56$,

(S2) $P(A'\cap B')=\dfrac1{18}$

Then

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If $\sigma^2$ is the variance of X, then $100\sigma^2$ is equal to ______.

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 :

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :

The probability distribution of a random variable X is $P(X=i)=\dfrac1{2^i}$, i = 1, 2, 3, ........ The variance of X is equal to _______.

Consider three sets $E_1 = \{1, 2, 3\}$, $F_1 = \{1, 3, 4\}$ and $G_1 = \{2, 3, 4, 5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements. Let $E_2 = E_1 - S_1$ and $F_2 = F_1 \cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements.

Let $G_2 = G_1 \cup S_2$. Finally, two elements are chosen at random, without replacement, from the set $G_2$ and let $S_3$ denote the set of these chosen elements.

Let $E_3 = E_2 \cup S_3$. Given that $E_1 = E_3$, let $p$ be the conditional probability of the event $S_1 = \{1, 2\}$. Then the value of $p$ 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 ____.

Let B$_i$ (i = 1, 2, 3) be three independent events in a sample space. The probability that only B$_1$ occur is α, only B$_2$ occurs is β and only B$_3$ occurs is γ. Let p be the probability that none of the events B$_i$ occurs and these 4 probabilities satisfy the equations (α − 2β) p = αβ and (β − 3γ) p = 2βγ (All the probabilities are assumed to lie in the interval (0, 1)). Then $\dfrac{P(B_1)}{P(B_3)}$ is equal to _______.

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\dfrac23$ and $\dfrac13$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2 - \alpha x + \beta$ are real and equal, 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 ______

There are three bags $B_1$, $B_2$ and $B_3$. The bag $B_1$ contains 5 red and 5 green balls, $B_2$ contains 3 red and 5 green balls, and $B_3$ contains 5 red and 3 green balls. Bags $B_1$, $B_2$ and $B_3$ have probabilities $\dfrac3{10}$, $\dfrac3{10}$ and $\dfrac4{10}$ respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Let $S$ be the sample space of all $3\times3$ matrices with entries from the set $\{0, 1\}$. Let the events $E_1$ and $E_2$ be given by

$E_1 = \{A\in S : \det A = 0\}$ and

$E_2 = \{A\in S : \text{sum of entries of } A \text{ is } 7\}$.

If a matrix is chosen at random from $S$, then the conditional probability $P(E_1|E_2)$ equals ______

Let $|X|$ denote the number of elements in a set $X$. Let $S = \{1, 2, 3, 4, 5, 6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$, then the number of ordered pairs $(A, B)$ such that $1 \le |B| < |A|$, equals ______