Total Probability Theorem and Bayes' Theorem

In a bolt factory, machines $X,Y,Z$ manufacture bolts that are respectively $25\%,35\%$ and $40\%$ of the factory's total output. The machines $X,Y,Z$ respectively produce $2\%,4\%$ and $5\%$ defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine $X$ ?

In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances of getting a particular chest disorder are $35\%,20\%$ and $10\%$ respectively. A person is chosen from the group at random and is found to be suffering from the chest disorder. The probability that the selected person is a smoker and non-vegetarian is :

Of cigarette smoking population $70\%$ are men and $30\%$ are women, $10\%$ of these men and $20\%$ of these women smoke Wills. The probability that a person seen smoking a Wills to be men is

A bag contains $20$ coins. If the probability that the bag contains exactly $4$ biased coin is $\frac{1}{3}$ and that of exactly $5$ biased coin is $\frac{2}{3}$, then the probability that all the biased coin are sorted out from the bag in exactly $10$ draws is

There are two urns. There are $m$ white & $n$ black balls in the first urn and $p$ white & $q$ black balls in the second urn. One ball is taken from the first urn & placed into the second. Now, the probability of drawing a white ball from the second urn is

In four schools $B_1\mathit{,}\mathit{ }{B}_{2 }\mathit{,}\mathit{ }{B}_3\mathit{,}\mathit{ }{B}_4$ the percentage of girls students is $12, 20, 13, 17$, respectively. From a school selected at random, one student is picked up at random, and it is found that the student is a girl. The probability that the school selected is $B_2$, is

Bag $A$ contains $4$ green and $3$ red balls and bag $B$ contains $4$ red and $3$ green balls. One bag is taken at random and a ball is drawn and noted it is green. The probability that it comes from bag $B$ is

There are two boxes. One box contains $3$ white balls and $2$ black balls. The other box contains $7$ yellow balls and $3$ black balls. If a box is selected at random and from it, a ball is drawn, the probability that the ball is black 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