There are $3$ bags which are known to contain $2$ white and $3$ black balls; $4$ white and $1$ black balls and $3$ white and $7$ black balls respectively. $A$ ball is drawn at random from one of the bags and found to be a black ball. Then the probability that it was drawn from the bag containing the most black balls is

$A$ computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20 \%$ and plant $T_2$ produces $80 \%$ of the total computers produced. $7 \%$ of computers produced in the factory turn out to be defective. It is known that $P(\text{defective} | T_1) = 10 P(\text{defective} | T_2)$. $A$ computer produced in the factory is randomly selected and it is not defective. Then the probability that it is produced in plant $T_2$ is

An urn contains $6$ white and $9$ black balls. Two successive draws of $4$ balls are made without replacement. The probability,that the first draw gives all white balls and the second draw gives all black balls,is:

$A$ bag contains $19$ unbiased coins and one coin with head on both sides. One coin is drawn at random and tossed,and a head turns up. If the probability that the drawn coin was unbiased is $\frac{m}{n}$,where $\operatorname{gcd}(m, n) = 1$,then $n^2 - m^2$ is equal to:

In a group of $400$ people,$160$ are smokers and non-vegetarian,$100$ are smokers and vegetarian,and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35\, \%, 20 \,\%$,and $10 \,\%$ respectively. $A$ person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian 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