Probability Tes...

Alice, Bob and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice, Carol always follows Bob, and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is 1/6, independent of the outcome of any other toss.)

A candidate takes three tests in succession and the probability of passing the first test is P. The probability of passing each succeeding test is P or $$\frac{P}{2}$$ according as he passes or fails in the preceding one. The candidate is selected if he passes at least two tests. The probability that the cndidate is selected is

A certain party consists of four different group of people - 30 students, 35 politicians, 20 actors and 27 leaders. On a particular function day, the total cost spent on party members was Rs. 9000. It was found that 6 students spent as much as 7 politicians, 15 politicians spent as much as 12 actors and 10 actors spent as much as 9 leaders. How much did students spent ?

Two events $$A$$ and $$B$$ are such that $$P(A)=\cfrac{1}{4},P(B)=\cfrac{1}{2}$$ and $$P(B| A)=\cfrac{1}{2}$$
Consider the following statements
$$(I)$$ $$P(\overline { A } |\overline { B } )=\cfrac { 3 }{ 4 } $$
$$(II)$$ $$A$$ and $$B$$ are mutually exclusive
$$(III)$$ $$P(\overline { A } |\overline { B } )+P(A|\overline { B } )=1$$
Then

A screw factory has two machines, the M1, which is old, and does $$75 \%$$ of all the screws, and the M2, newer but small, that does $$25 \%$$ of the screws. The M1 does $$4 \%$$ of defective screws, while the M2 just does $$2 \%$$ of defective screws. If we choose a screw at random: what is the probability that it turns out to be defective?

We roll a fair four-sided die. If the result is $$1$$ or $$2$$, we roll once more but otherwise, we stop. What is the probability that the sum total of our rolls is at least $$4$$?

A business man is expecting two telephone calls. Mr Walia may call any time between $$2$$ p.m and $$4$$ p.m. while Mr Sharma is equally likely to call any time between $$2.30$$ p.m. and $$3.15$$ p.m. The probability that Mr Walia calls before Mr Sharma is:

A factory production line is manufacturing bolts using three machines, A, B and C. Of the total output, machine A is responsible for $$25$$%, machine B for $$35$$% and machine C for the rest. It is known from previous experience with the machines that $$5$$% of the output from machine A is defective, $$4$$% from machine B and $$2$$% from machine C. A bolt is chosen at random from the production line and found to be defective. What is the probability that it came from machine C? 

In a group $$14$$ males and $$6$$ females, $$8$$ and $$3$$ of the males and females respectively are aged above $$40$$ years. The probability that a person selected at random from the group is aged above $$40$$ years, given that the selected person is female, is

A student answers a multiple choice question with $$5$$ alternatives, of which exactly one is correct. The probability that he knows the correct answer is $$p,0< p< 1$$. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly, is