For two given events $A$ and $B$, $P\,(A \cap B) = $

Let $A$ and $B$ be two events such that $P\overline {(A \cup B)} = \frac{1}{6},P(A \cap B) = \frac{1}{4}$ and $P(\bar A) = \frac{1}{4},$ where $\bar A$ stands for complement of event $A$. Then events $A$ and $B$ are

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 )+ $ $P ( A \cap B \cap C )$

Two cards are drawn at random and without replacement from a pack of $52$ playing cards. Finds the probability that both the cards are black.

An integer is chosen at random from the integers $\{1,2,3, \ldots \ldots . .50\}$. The probability that the chosen integer is a multiple of atleast one of $4,6$ and $7$ is

A card is drawn from a pack of $52$ cards. A gambler bets that it is a spade or an ace. What are the odds against his winning this bet