$A$ bag contains $6$ balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least $5$ black balls is

$A$ factory has a total of three manufacturing units,$M_1, M_2$,and $M_3$,which produce bulbs independently. The units $M_1, M_2$,and $M_3$ produce bulbs in the proportions of $2: 2: 1$,respectively. It is known that $20\%$ of the bulbs produced in the factory are defective. It is also known that,of all the bulbs produced by $M_1, 15\%$ are defective. Suppose that,if a randomly chosen bulb produced in the factory is found to be defective,the probability that it was produced by $M_2$ is $\frac{2}{5}$. If a bulb is chosen randomly from the bulbs produced by $M_3$,then the probability that it is defective is $.....$ .

$A$ man is known to speak the truth $4$ out of $5$ times. He throws a die and reports that it is a six. The probability that it was actually a six is

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

In answering a question on a multiple choice test,a student either knows the answer or guesses. Let $\frac{3}{4}$ be the probability that he knows the answer and $\frac{1}{4}$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac{1}{4}$. What is the probability that the student knows the answer given that he answered it correctly?

Bag $A$ contains $2$ white,$1$ black and $3$ red balls and bag $B$ contains $3$ black,$2$ red and $n$ white balls. One bag is chosen at random and $2$ balls drawn from it at random,are found to be $1$ red and $1$ black. If the probability that both balls come from Bag $A$ is $\frac{6}{11}$,then $n$ is equal to