Probability Tes...

$$A$$ is one of $$6$$ horses entered for a race, and is to be ridden by one of two jockeys $$B$$ and $$C$$. It is $$2$$ to $$1$$ that $$B$$ rides $$A$$, in which case all the horses are equally likely to win; if $$C$$ rides $$A$$, his chance is trebled; what are the odds against his winning?

An unbiased coin is tossed. if the result is a head, a pair of unbiased dice is rolled & the number obtained by adding the numbers on the two faces is noted. If the result is a tail, a card from a well shuffled pack of eleven cards numbered  $$2, 3, 4,...., 12$$  is picked & the number on the card is noted. What is the probability that the noted number is either $$7$$  or $$8 $$?

There are two groups of subjects one of which consists of 5 science subjects and 3 engineering subjects and the other consists of 3 science and 5 engineering subjects. An unbaised die is cast. If number 3 or number 5 turns up, a subject is selected at random from the first group, other wise the subject is selected at random from the second group. Find the probability that an engineering subject is selected ultimately.

A lot of contains $$20$$ articles. The probability that the lot contains exactly $$2$$ defective articles is $$0.4$$ and the probability that the lot contains exactly $$3$$ defective articles is $$0.6$$. Articles are drawn from the lot at random one by one without replacement and are tested till all defective articles are found. What is the probability that the testing procedure ends at the twelfth testing?

There are $$3$$ bags each containing $$5$$ white balls and $$2$$ black balls and $$2$$ bags each containing $$1$$ white balls and $$4$$ black balls, a black ball having been drawn, find the chance that it came from the first group.

There are two balls in an urn whose colours are not known (each ball can be either white or black). A white ball is put into the urn. A ball is drawn from the urn. The probability that it is white is

A bag contains some white and some black balls, all combinations of balls 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

A bag contains some white and some black balls, all combinations of balls 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

A letter is known to have come from either $$TATANAGAR$$ or $$CALCUTTA$$. On the envelope just two consecutive letters $$TA$$ are visible. The probability that the letter has come from $$CALCUTTA$$ 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