$A$ bag $P$ contains $4$ red and $5$ black balls,another bag $Q$ contains $3$ red and $6$ black balls. If one ball is drawn at random from bag $P$ and two balls are drawn from bag $Q$,then the probability that out of the three balls drawn two are black and one is red,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 $A$ be the set of all $4$-digit natural numbers whose exactly one digit is $7$. Then the probability that a randomly chosen element of $A$ leaves a remainder of $2$ when divided by $5$ is ..... .

Three dice are thrown simultaneously and the sum of the numbers appeared on them is noted. If $A$ is the event of getting a sum greater than $14$ and $B$ is the event of getting a sum which is a multiple of $3$,then $P(A \cap \overline{B}) + P(\overline{A} \cap B) = $

Cards are drawn one by one without replacement from a pack of $52$ cards. The probability that $10$ cards will precede the first ace is

$A$ locker can be opened by dialing a fixed three-digit code (between $000$ and $999$). $A$ stranger who does not know the code tries to open the locker by dialing three digits at random. The probability that the stranger succeeds at the $k^{th}$ trial is