Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

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

Out of all possible $8$ digit numbers formed using all the digits $0,0,1,1,2,3,4,4$ a number is randomly selected. Probability that the selected number is odd, is-

A box contains $10$ red marbles, $20$ blue marbles and $30$ green marbles. $5$ marbles are drawn from the box, what is the probability that atleast one will be green?

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :

Among $15$ players, $8$ are batsmen and $7$ are bowlers. Find the probability that a team is chosen of $6$ batsmen and $5$ bowlers