$A, B, C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting the target are $\frac{3}{4}, \frac{1}{2}, \frac{5}{8}$. The probability that the target is hit by $A$ or $B$ but not by $C$ is

In a battery manufacturing factory, machines $P$, $Q$ and $R$ manufacture $20 \%$, $30 \%$ and $50 \%$ respectively of the total output. The chances that a defective battery is produced by these machines are $1 \%$, $1.5 \%$ and $2 \%$ respectively. If a battery is selected at random from the production, then the probability that it is defective is

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. $A$ fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die,then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

An urn $A$ contains $4$ white and $1$ black ball; urn $B$ contains $3$ white and $2$ black balls and urn $C$ contains $2$ white and $3$ black balls. One ball is transferred randomly from $A$ to $B$; later one ball is transferred randomly from $B$ to $C$. Finally,if a ball is drawn randomly from $C$,then the probability that it is a black ball is

Cards are drawn one after the other without replacement from a well-shuffled pack of cards until an ace card appears. If the probability that exactly $5$ cards are drawn before the first ace card appears is $\frac{4}{49}\left(\frac{p_1 \cdot p_2 \cdot p_3}{p_4 \cdot p_5 \cdot p_6}\right)$, where $p_i$ is prime for $i=1, 2, 3, 4, 5, 6$, then $(\max \{p_i\} - \min \{p_i\}) = $

Let $p$ denote the probability that a man aged $x$ years will die in a year. The probability that out of $n$ men $A_1, A_2, A_3, ..., A_n$ each aged $x$,$A_1$ will die in a year and will be the first to die,is