Probability Tes...

A bag contains some white and some black balls, all of which are distinguishable from each other, all combinations of balls being equally likely. The total number of balls in the bag is $$10$$. If three balls are drawn at random and all of them are found to be black, the probability that the bag contains $$1$$ white and $$9$$ black balls is :

A man is known to speak the truth $$3$$ out of $$4$$ times. He throws a die and reports that it is a six. The probability that it is actually a six is

In an entrance test, there are multiple choice questions. There are four possible options of which one is correct. The probability that a student knows the answer to a question is $$90$$%. If he gets the correct answer to a question, then the probability that he was guessing is

An artillery target may be either at point $$I$$ with probability $$\cfrac{8}{9}$$ or at point $$II$$ with probability $$\cfrac{1}{9}$$. We have $$21$$ shells each of which can be fired at point $$I$$ or $$II$$. Each shell may hit the target independently of the other shell with probability $$\cfrac{1}{2}$$. How many shells must be fired at point $$I$$ to hit the target with maximum probability?

A bag $$A$$ contains $$2$$ white and $$3$$ red balls, and a bag $$B$$ contains $$4$$ white and $$5$$ red balls. One ball is drawn at random from one of the bags and it is found to be red. Find the the probability that it was drawn from the bag $$B$$.

In a bolt factory, machines $$A,B$$ and $$C$$ manufacture $$25$$%, $$35$$%, $$40$$% respectively. Of the total of their output $$5$$%,$$4$$% and $$2$$% are defective. A bolt is drawn and is found to be defective. What are the probabilities that it was manufactured by the machines $$A,B,C$$.

A bag contains some white and some black balls, all combinations of ball being equally likely. The total number of balls in the bag is 10. If three balls are drawn at random without replacement and all of them are found to be black, the probability that the bag contains 1 white and 9 black balls is

If a selected student has been found to pass the examination then find out the probability that he is the only student to have passed the examination.

In a test an examinee either guesses or copies or knows the answer to a multiple choice question with  $$4$$ choices. The probability that he makes a guess is $$\dfrac{1}{3}$$ and the probability that he copies the answer is $$\dfrac{1}{6}$$. The probability that his answer is correct given that he copied it, is $$\dfrac{1}{8}$$. Find the probability that he knew the answer to the question given that he correctly answered it.

Die $$A$$ has $$4$$ red and $$2$$ white faces where as die $$B$$ has two red and $$4$$ white faces. $$A$$ fair coin is tossed. If head turns up, the game continues by throwing die $$A$$, if tail turns up then die $$B$$ is to be used. If the first two throws resulted in red, what is the probability of getting red face at the third throw ?