For independent events $A$ and $B$,if $P(A) = \frac{1}{2}$ and $P(A \cup B) = \frac{3}{5}$,then $P(B) =$ . . . . . . .

An urn $A$ contains $4$ white and $1$ black ball; urn $B$ contains $3$ white and $2$ black balls and urn $C$ contains $2$ white and $3$ black balls. One ball is transferred randomly from $A$ to $B$; later one ball is transferred randomly from $B$ to $C$. Finally,if a ball is drawn randomly from $C$,then the probability that it is a black ball is

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are the same. If the probability of a random toss resulting in a head is $\frac{1}{3}$,then the probability that the experiment stops with heads is:

If $A$ and $B$ are two events of a random experiment such that $P(\bar{A})=\frac{2}{3}$, $P(B)=\frac{4}{15}$ and $P(A \cap \bar{B})=\frac{1}{5}$, then $\sqrt{195[P(B \mid(A \cup \bar{B}))+P(A \cup B)]} = $

Two persons $A$ and $B$ throw a fair die (six-faced cube with faces numbered from $1$ to $6$) alternately,starting with $A$. The first person to get an outcome different from the previous one thrown by the opponent wins. The probability that $B$ wins is:

$A$ 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?