If $A_i$ $(i=1, 2, 3, \ldots, n)$ are $n$ independent events with $P(A_i) = \frac{1}{1+i}$ for each $i$,then the probability that none of $A_i$ occurs is

$A$ purse contains $4$ copper coins and $3$ silver coins,the second purse contains $6$ copper coins and $2$ silver coins. If a coin is drawn out of one of these purses,then the probability that it is a copper coin is :-

Two marbles are drawn in succession from a box containing $10$ red,$30$ white,$20$ blue and $15$ orange marbles,with replacement being made after each drawing. Then the probability,that the first drawn marble is red and the second drawn marble is white,is

$A$ bag $X$ contains $2$ white and $3$ black balls,and another bag $Y$ contains $4$ white and $2$ black balls. If one bag is selected at random and one ball is drawn from it,what is the probability that the ball is white?

Events $A$ and $B$ are independent events and $P(A)=P$,$P(B)=\frac{1}{2}$ and $P(A \cup B)=\frac{3}{5}$,then the value of $P$ is . . . . . . .

$A$ bag $x$ contains $3$ white balls and $2$ black balls and another bag $y$ contains $2$ white balls and $4$ black balls. $A$ bag and a ball out of it are picked at random. The probability that the ball is white,is