If $P(A)=\frac{6}{11}, P(B)=\frac{5}{11}$ and $P(A \cup B)=\frac{7}{11},$ find $P(A \cap B).$

If $A$ and $B$ are two events in a random experiment such that $P(A) + P(B) = 2 P(A \cap B)$, then:

$A$ party of $23$ persons take their seats at a round table. The odds against two specific persons sitting together are

In class $XI$ of a school,$40\%$ of the students study Mathematics and $30\%$ study Biology. $10\%$ of the class study both Mathematics and Biology. If a student is selected at random from the class,find the probability that they will be studying Mathematics or Biology.

$A$ card is drawn from a pack of $52$ cards. Find the probability that the card will be a queen or a heart.

$A$ die is thrown. Let $A$ be the event that the number obtained is greater than $3$. Let $B$ be the event that the number obtained is less than $5$. Then $P(A \cup B)$ is