The probability that at least one of the events $E_1$ and $E_2$ occurs is $0.6$. If the probability of the simultaneous occurrence of $E_1$ and $E_2$ is $0.2$,then $P(E_1') + P(E_2') = $

If a number is chosen at random from the first $200$ positive integers,what is the probability that it is divisible by $6$ or $8$?

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

An electronic assembly consists of two subsystems,$A$ and $B$. From previous testing procedures,the following probabilities are known:
$P(A \text{ fails}) = 0.2$
$P(B \text{ fails alone}) = 0.15$
$P(A \text{ and } B \text{ fail}) = 0.15$
Evaluate the probability $P(A \text{ fails alone})$.

Let $A = \{1, 6, 11, 16, \dots\}$ and $B = \{9, 16, 23, 30, \dots\}$ be the sets consisting of the first $2025$ terms of two arithmetic progressions. Then $n(A \cup B)$ is

Let $S$ be the set of all ordered pairs $(x, y)$ of positive integers satisfying the condition $x^2 - y^2 = 12345678$. Then,