JEE Advanced Paper 1 - 2013

Let $\left|X\right|$ denote the number of elements in set $X.$ Let $S=\left\{1, 2, 3, 4, 5, 6\right\}$ 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 $\left(A, B\right)$ such that $1\le \left|B\right|<\left|A\right|,$ equals

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

There are three bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3.$ The bag ${\mathrm{B}}_1$ contains $5$ red and $5$ green balls, ${\mathrm{B}}_2$ contains $3$ red and $5$ green balls, and ${\mathrm{B}}_3$ contains $5$ red and $3$ green balls, Bags ${\mathrm{B}}_1,{\mathrm{B}}_2$ and ${\mathrm{B}}_3$ have probabilities $\frac{3}{10},\frac{3}{10}$ and $\frac{4}{10}$ 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?

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

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

Let $E, F$ and $G$ be three events having probabilities $P\left(E\right)=\frac{1}{8},P\left(F\right)=\frac{1}{6}$ and $P\left(G\right)=\frac{1}{4},$ and let $P(E\cap F\cap G)=\frac{1}{10}$

For any event $H$, if $H^c$ denotes its complement, then which of the following statements is (are) TRUE ?

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$ and $\frac{1}{3}$, 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