Four boxes $A, B, C$ and $D$ contain $5000, 3000, 2000$ and $1000$ fuses respectively. The percentages of defective fuses in these boxes are $3\%, 2\%, 1\%$ and $0.5\%$ respectively. If a fuse selected at random from one of the boxes is found to be defective,then the probability that it has come from box $D$ is

$A$ signal which can be green or red with probability $\frac{4}{5}$ and $\frac{1}{5}$ respectively,is received by station $A$ and then transmitted to station $B$. The probability of each station receiving the signal correctly is $\frac{3}{4}$. If the signal received at station $B$ is green,then the probability that the original signal was green is

In a test,a student either guesses,copies,or knows the answer to a multiple-choice question with four choices having one correct answer. The probability that he guesses the answer is $\frac{1}{3}$ and the probability that he copies it is $\frac{1}{12}$. The probability that his answer is correct given that he copied it is $\frac{1}{6}$. The probability that he knew the answer,given that he has correctly answered it,is

$A$ bag contains $5$ balls of unknown colours. There are equal chances that out of these five balls,there may be $0, 1, 2, 3, 4,$ or $5$ red balls. $A$ ball is taken out from the bag at random and is found to be red. The probability that it is the only red ball in the bag is

An unbiased coin is tossed. If the outcome is a head,then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in a tail,then a card from a well-shuffled pack of nine cards numbered $1, 2, 3, \dots, 9$ is randomly picked and the number on the card is noted. The probability that the noted number is either $7$ or $8$ is

Bag $1$ contains $4$ white balls and $5$ black balls,and Bag $2$ contains $n$ white balls and $3$ black balls. One ball is drawn randomly from Bag $1$ and transferred to Bag $2$. $A$ ball is then drawn randomly from Bag $2$. If the probability that the ball drawn from Bag $2$ is white is $29/45$,then $n$ is equal to: