$A$ and $B$ are independent events with $P(A)=\frac{3}{10}$ and $P(B)=\frac{2}{5}$. Then,the value of $P(A^{\prime} \cup B)$ is:

Let $B_{i} (i=1, 2, 3)$ be three independent events in a sample space. The probability that only $B_{1}$ occurs is $\alpha$,only $B_{2}$ occurs is $\beta$,and only $B_{3}$ occurs is $\gamma$. Let $p$ be the probability that none of the events $B_{i}$ occurs,and these $4$ probabilities satisfy the equations $(\alpha - 2\beta)p = \alpha\beta$ and $(\beta - 3\gamma)p = 2\beta\gamma$ (All the probabilities are assumed to lie in the interval $(0, 1)$). Then $\frac{P(B_{1})}{P(B_{3})}$ is equal to ..........

In a game,a person wins $5$ rupees for getting a number greater than $4$ and loses $1$ rupee otherwise,when a fair die is thrown. $A$ man participates in the game and decides to quit as soon as he gets a number greater than $4$. If he plays for a maximum of $3$ throws,what is the expected value (mean value) of the amount he wins or loses?

If $7$ dice are thrown simultaneously,then the probability that all six digits appear on the upper faces is equal to -

An ordinary cube has four blank faces,one face marked $2$,and another marked $3$. The probability of obtaining a total of exactly $12$ in $5$ throws is:

If two numbers $p$ and $q$ are chosen randomly from the set $\{1, 2, 3, 4\}$,one by one,with replacement,then the probability of getting $p^2 > 4q$ is