An urn contains $7$ red,$5$ white and $3$ black balls. Three balls are drawn randomly one after the other without replacement. If it is known that the first ball drawn is red and the second ball drawn is white,then the probability that the third ball drawn is not red is

$A$ bag contains $10$ balls out of which $k$ are red and $(10-k)$ are black,where $0 \le k \le 10$. If three balls are drawn at random without replacement and all of them are found to be black,then the probability that the bag contains $1$ red and $9$ black balls is:

$A$ box contains $4$ black,$2$ white,and $6$ red balls. Another box contains $3$ black and $5$ white balls. An unbiased die is thrown. If $1$ or $2$ appears on the die,a ball is drawn from the first box; otherwise,a ball is drawn from the second box. If the drawn ball is black,what is the probability that $2$ appeared on the die?

Box $I$ contains $30$ cards numbered $1$ to $30$ and Box $II$ contains $20$ cards numbered $31$ to $50$. $A$ box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box $I$ is

Bag $A$ contains $6$ Green and $8$ Red balls and bag $B$ contains $9$ Green and $5$ Red balls. $A$ card is drawn at random from a well-shuffled pack of $52$ playing cards. If it is a spade,two balls are drawn at random from bag $A$,otherwise two balls are drawn at random from bag $B$. If the two balls drawn are found to be of the same colour,then the probability that they are drawn from bag $A$ 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