$A$ and $B$ are independent events with $P(A)=\frac{1}{4}$ and $P(A \cup B)=2 P(B)-P(A)$,then $P(B)$ is

Let $A$ and $B$ be independent events such that $P(A)=p$ and $P(B)=2p$. The largest value of $p$, for which $P(\text{exactly one of } A, B \text{ occurs}) = \frac{5}{9}$, is:

Consider a system of equations $ax + by = 0$ and $cx + dy = 0$,where $a, b, c, d \in \{0, 1\}$.
Statement $-1$: The probability that the system of equations has a solution is $1$.
Statement $-2$: The probability that the system of equations has a unique solution is $\frac{3}{8}$.

For three events $A$, $B$, and $C$ of a sample space, $P(\text{exactly one of } A \text{ or } B \text{ occurs}) = P(\text{exactly one of } B \text{ or } C \text{ occurs}) = P(\text{exactly one of } C \text{ or } A \text{ occurs}) = \frac{1}{4}$. If the probability of all the three events occurring simultaneously is $\frac{1}{16}$, then the probability that at least one of the events occurs is:

$P$ speaks truth in $70\%$ of the cases and $Q$ in $80\%$ of the cases. In what percent of cases are they likely to agree in stating the same fact (in $\%$)?

For three events $A, B$ and $C$,$P(\text{Exactly one of } A \text{ or } B \text{ occurs}) = P(\text{Exactly one of } B \text{ or } C \text{ occurs}) = P(\text{Exactly one of } C \text{ or } A \text{ occurs}) = \frac{1}{4}$ and $P(\text{All the three events occur simultaneously}) = \frac{1}{16}$. Then the probability that at least one of the events occurs is: