Independent events

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let $E_1$ be the event that the ball chosen belonged to Box I and let $E_2$ be the event that the ball chosen belonged to Box II. Let $F_1$ be the event that the ball chosen is red and let $F_2$ be the event that the ball chosen is green.

Then which of the following statements is (are) TRUE ?

Three students $S_1, S_2$, and $S_3$ are given a problem to solve. Consider the following events:

• $U$: At least one of $S_1, S_2$, and $S_3$ can solve the problem,
• $V$: $S_1$ can solve the problem, given that neither $S_2$ nor $S_3$ can solve the problem,
• $W$: $S_2$ can solve the problem and $S_3$ cannot solve the problem,
• $T$: $S_3$ can solve the problem.

For any event $E$, let $P(E)$ denote the probability of $E$. If

$$P(U) = \frac12,\qquad P(V) = \frac{1}{10},\qquad \text{and}\qquad P(W) = \frac{1}{12},$$

then $P(T)$ is equal to

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is $\dfrac13$, then the probability that the experiment stops with head is

Two players, $P_1$ and $P_2$, play a game against each other. In every round of the game, each player rolls a fair die once, where the six faces of the die have six distinct numbers. Let $x$ and $y$ denote the readings on the die rolled by $P_1$ and $P_2$, respectively. If $x > y$, then $P_1$ scores 5 points and $P_2$ scores 0 point. If $x = y$, then each player scores 2 points. If $x < y$, then $P_1$ scores 0 point and $P_2$ scores 5 points. Let $X_i$ and $Y_i$ be the total scores of $P_1$ and $P_2$, respectively, after playing the $i^{th}$ round.

The correct option is:

Let $E_1$ and $E_2$ be two events such that the conditional probabilities $P(E_1|E_2)=\dfrac12$, $P(E_2|E_1)=\dfrac34$ and $P(E_1\cap E_2)=\dfrac18$. Then :

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark $n$ is $\dfrac1n$. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, 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 ______

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 ______

Three persons P, Q and R independently try to hit a target. If the probabilities of their hitting the target are $\dfrac34$, $\dfrac12$ and $\dfrac58$ respectively, then the probability that the target is hit by P or Q but not by R is :

Let E and F be two independent events. The probability that both E and F happen is $\dfrac1{12}$ and the probability that neither E nor F happens is $\dfrac12$, then a value of $\dfrac{P(E)}{P(F)}$ is :

$P(X>Y)$ is

$P(X=Y)$ is

Let two fair six-faced dice A and B be thrown simultaneously. If $E_1$ is the event that die A shows up four, $E_2$ is the event that die B shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is NOT true ?

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

Let A and B be two events such that $P\left(\overline{A\cup B}\right)=\dfrac16$, $P(A\cap B)=\dfrac14$ and $P\left(\bar A\right)=\dfrac14$, where $\bar A$ stands for the complement of the event A. Then the events A and B are :

Four persons independently solve a certain problem correctly with probabilities $\frac12,\frac34,\frac14,\frac18$. Then the probability that the problem is solved correctly by at least one of them is

Of the three independent events $E_1$, $E_2$ and $E_3$, the probability that only $E_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$ and only $E_3$ occurs is $\gamma$. Let the probability $p$ that none of events $E_1$, $E_2$ or $E_3$ occurs satisfy the equations $(\alpha-2\beta)p=\alpha\beta$ and $(\beta-3\gamma)p=2\beta\gamma$. All the given probabilities are assumed to lie in the interval $(0,1)$.

Then $\dfrac{\text{Probability of occurrence of }E_1}{\text{Probability of occurrence of }E_3}=$

If 1 ball is drawn from each of the boxes $B_1$, $B_2$ and $B_3$, the probability that all 3 drawn balls are of the same colour is