$A$ and $B$ are two independent events such that $P(A \cup B) = 0.8$ and $P(A) = 0.3$. The value of $P(B)$ is:

If $A$ and $B$ are two independent events such that $P(A) > 0.5$,$P(B) > 0.5$,$P(A \cap \bar{B}) = \frac{3}{25}$,and $P(\bar{A} \cap B) = \frac{8}{25}$,then $P(A \cap B)$ is:

In a game,a person wins $5$ rupees for getting a number greater than $4$ and loses $1$ rupee otherwise,when a fair die is thrown. $A$ man participates in the game and decides to quit as soon as he gets a number greater than $4$. If he plays for a maximum of $3$ throws,what is the expected value (mean value) of the amount he wins or loses?

If $A, B$ and $C$ are three independent events such that $P(A)=P(B)=P(C)=P$, then $P$ (at least two of $A, B$ and $C$ occur) is equal to

Let $p$ denote the probability that a man aged $x$ years will die in a year. The probability that out of $n$ men $A_1, A_2, A_3, ..., A_n$ each aged $x$,$A_1$ will die in a year and will be the first to die,is

Let $X$ and $Y$ be events such that $P(X \cup Y) = P(X \cap Y).$
Statement-$1$: $P(X \cap Y) = P(X' \cap Y') = 0$
Statement-$2$: $P(X) + P(Y) = 2P(X \cap Y).$