Two numbers $x$ and $y$ are chosen at random from the set of integers $\{1,2,3,4......15\}.$ The probability that point $(x,y)$ lies on a line through $(0,0)$ having slope $\frac{2}{3}$ is

Dialing a telephone number an old man forgets the last two digits remembering only that these are different dialled at random. The probability that the number is dialled correctly, is

The probability of getting $4$ heads in $8$ throws of a coin, is

A cricket team has $15$ members, of whom only $5$ can bowl. If the names of the $15$ members are put into a hat and $11$ drawn at random, then the chance of obtaining an eleven containing at least $3$ bowlers is

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 probability that the roots of the quadratic polynomial $x^2-\alpha x+\beta$ are real and equal, is

Mr. $A$ has six children and atleast one child is a girl, then probability that Mr. $A$ has $3$ boys and $3$ girls, is -