Box $A$ contains $2$ black and $3$ red balls,while Box $B$ contains $3$ black and $4$ red balls. Out of these two boxes,one is selected at random; and the probability of choosing Box $A$ is double that of Box $B$. If a red ball is drawn from the selected box,then the probability that it has come from Box $B$ is:

In an examination,there are $4$ Yes/No type questions. The probability that a student answers a question correctly without guessing is $2/3$. The probability that a student guesses a correct answer is $1/2$. $A$ student writes the examination either by not guessing any of the $4$ questions or by guessing all $4$ questions. The probability that they attempt the exam by guessing all questions is $3/7$. Given that a student answered at least $3$ questions correctly,what is the probability that they answered all questions without guessing?

$A$ pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade 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

Two balls are selected at random one by one without replacement from a bag containing $4$ white and $6$ black balls. If the probability that the first selected ball is black,given that the second selected ball is also black,is $\frac{m}{n}$,where $\operatorname{gcd}(m, n) = 1$,then $m + n$ is equal to :

The probabilities of having a defective toy in three cartons $A, B, C$ are $\frac{1}{3}, \frac{1}{4}, \frac{2}{5}$ respectively. If a carton is selected at random and a toy drawn randomly from it is found to be defective,then the probability that it is drawn from carton $B$ is