$A$ six-faced die is biased such that it is twice as likely to show an even number as an odd number when thrown. If the die is thrown twice,what is the probability that the sum of the two numbers obtained is even?

If $E_1, E_2, \ldots, E_n$ are independent events such that $P(E_r) = \frac{1}{1+r}$ for $r = 1, 2, \ldots, n$,then the probability that at least one of $E_1, E_2, \ldots, E_n$ happens is

Two persons $A$ and $B$ take turns in throwing a pair of dice. The first person to throw a sum of $9$ with both dice will win the game. If $A$ throws first,then the probability that $B$ wins the game is:

In a tournament,there are $12$ players $P_1, P_2, P_3, \dots, P_{12}$ divided into $6$ pairs at random. From each game,a winner is decided based on the game played between the two players of the pair. Assuming each player is of equal strength,what is the probability that exactly one out of $P_1$ and $P_2$ is among the losers?

$A$ pair of fair dice is rolled together till a sum of either $5$ or $7$ is obtained. Then the probability that $5$ comes before $7$ is

$A$ and $B$ are independent events. If $P(A \cup B)=0.5$ and $P(A)=0.2$,then $P(B) = $ . . . . . . . (in $/8$)