Let $A, B,$ and $C$ be events such that $P(A) = P(B) = P(C) = \frac{1}{4}, P(AB) = P(CB) = 0,$ and $P(AC) = \frac{1}{8}.$ Then $P(A \cup B) = \dots$

If $A, B, C$ are three events associated with a random experiment,prove that $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)$.

If $A$ and $B$ are two events such that $P(A \cup B) + P(A \cap B) = \frac{7}{8}$ and $P(A) = 2P(B)$,then $P(A) = $

The probability that $A$ speaks the truth is $\frac{4}{5}$,while the probability that $B$ speaks the truth is $\frac{3}{4}$. What is the probability that they contradict each other when asked to speak on a fact?

For any two events $A$ and $B$,if $P(A \cup B) = a P(A \cap B) + b P(A) + c P(B)$,then $3a + 2b + 5c = ?$

Let $A$ and $B$ be two independent events of an experiment. If $P(A) = 0.3$ and $P(A \cup B) = 0.8$,then find $P(A \to B)$,where $P(X)$ denotes the probability that statement $X$ is true.