Probability

If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is :

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is

The probability that $x_1+x_2+x_3$ is odd, is

The probability that $x_1$, $x_2$, $x_3$ are in an arithmetic progression, 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 :

If A and B are two events such that $P(A\cup B)=P(A\cap B)$, then the incorrect statement amongst the following statements is :

A set S contains 7 elements. A non-empty subset A of S and an element $x$ of S are chosen at random. Then the probability that $x\in A$ is :

A number $x$ is chosen at random from the set {1, 2, 3, 4, ....., 100}. Define the event : A = the chosen number $x$ satisfies
$$\frac{(x-10)(x-50)}{(x-30)}\ge0$$
Then P(A) is :

Let A and E be any two events with positive probabilities :

Statement - 1 : $P(E/A)\ge P(A/E)P(E)$

Statement - 2 : $P(A/E)\ge P(A\cap E)$.

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

If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the other ball is red, the probability that these 2 balls are drawn from box $B_2$ is

A ship is fitted with three engines $E_1$, $E_2$ and $E_3$. The engines function independently of each other with respective probabilities $\frac12$, $\frac14$ and $\frac14$. For the ship to be operational at least two of its engines must function. Let $X$ denote the event that the ship is operational and let $X_1$, $X_2$ and $X_3$ denote respectively the events that the engines $E_1$, $E_2$ and $E_3$ are functioning. Which of the following is (are) true?

Four fair dice $D_1$, $D_2$, $D_3$ and $D_4$, each having six faces numbered 1, 2, 3, 4, 5 and 6, are rolled simultaneously. The probability that $D_4$ shows a number appearing on one of $D_1$, $D_2$ and $D_3$ is

Let $X$ and $Y$ be two events such that $P(X|Y)=\dfrac12$, $P(Y|X)=\dfrac13$ and $P(X\cap Y)=\dfrac16$. Which of the following is (are) correct?

The probability of the drawn ball from $U_2$ being white is

Given that the drawn ball from $U_2$ is white, the probability that head appeared on the coin is

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\dfrac{11}{25}$ and the probability of none of them occurring is $\dfrac2{25}$. If $P(T)$ denotes the probability of occurrence of the event $T$, then

Let $\omega$ be a complex cube root of unity with $\omega\ne1$. A fair die is thrown three times. If $r_1$, $r_2$ and $r_3$ are the numbers obtained on the die, then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is