There are $2$ bags each containing $3$ white and $5$ black balls and $4$ bags each containing $6$ white and $4$ black balls. If a ball drawn randomly from a bag is found to be black,then the probability that this ball is from the first set of bags is

Two groups are competing for the position on the board of directors of a corporation. The probabilities that the first and the second groups will win are $0.6$ and $0.4$ respectively. Further,if the first group wins,the probability of introducing a new product is $0.7$ and the corresponding probability is $0.3$ if the second group wins. Find the probability that the new product was introduced by the second group. (in $/9$)

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

$A, B_1, B_2, B_3$ are the events in a random experiment. If $P(B_1)=0.25, P(B_2)=0.30, P(B_3)=0.45, P(A|B_1)=0.05, P(A|B_2)=0.04, P(A|B_3)=0.03$, then $P(B_2|A) = $

$A$ company produces $10,000$ items per day. On a particular day, $2500$ items were produced on machine $A$, $3500$ on machine $B$, and $4000$ on machine $C$. The probability that an item produced by machines $A, B, C$ is defective is $2 \%$, $3 \%$, and $5 \%$, respectively. If one item is selected at random from the output and is found to be defective, then the probability that it was produced by machine $C$ is:

In an entrance test,there are multiple-choice questions. There are four possible answers to each question,of which one is correct. The probability that a student knows the answer to a question is $9/10$. If he gets the correct answer to a question,then the probability that he was guessing is: