In a class of $125$ students $70$ passed in Mathematics, $55$ in Statistics and $30$ in both. The probability that a student selected at random from the class has passed in only one subject is

If $P(A) = P(B) = x$ and $P(A \cap B) = P(A' \cap B') = \frac{1}{3}$, then $x = $

Let $A$ and $B$ be independent events with $P(A)=0.3$ and $P(B)=0.4$. Find $P(A \cup B)$

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

A die marked $1,\,2,\,3$ in red and $4,\,5,\,6$ in green is tossed. Let $A$ be the event, $'$ the number is even,$'$ and $B$ be the event, 'the number is red'. Are $A$ and $B$ independent?

If $A$ and $B$ are two independent events, then the probability of occurrence of at least one of $\mathrm{A}$ and $\mathrm{B}$ is given by $1 -\mathrm{P}\left(\mathrm{A}^{\prime}\right) \mathrm{P}\left(\mathrm{B}^{\prime}\right)$