Three companies $C_1, C_2, C_3$ produce car tyres. $A$ car manufacturing company buys $40 \%$ of its requirement from $C_1, 35 \%$ from $C_2$ and $25 \%$ from $C_3$. The company knows that $2 \%$ of the tyres supplied by $C_1, 3 \%$ by $C_2$ and $4 \%$ by $C_3$ are defective. If a tyre chosen at random from the consignment received is found defective, then the probability that it was supplied by $C_2$ is:

An instructor has a question bank consisting of $300$ easy True / False questions,$200$ difficult True / False questions,$500$ easy multiple choice questions,and $400$ difficult multiple choice questions. If a question is selected at random from the question bank,what is the probability that it will be an easy question given that it is a multiple choice question?

$25 \%$ of the population are smokers. $A$ smoker has $27$ times more chances to develop lung cancer than a non-smoker. $A$ person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of $k$ is $.............$

$A$ doctor is to visit a patient. From past experience,it is known that the probabilities that he will come by train,bus,scooter,or by other means of transport are respectively $\frac{3}{10}, \frac{1}{5}, \frac{1}{10},$ and $\frac{2}{5}$. The probabilities that he will be late are $\frac{1}{4}, \frac{1}{3},$ and $\frac{1}{12}$ if he comes by train,bus,and scooter respectively,but if he comes by other means of transport,then he will not be late. When he arrives,he is late. What is the probability that he comes by train?

$A$ box $A$ contains $2$ white,$3$ red and $2$ black balls. Another box $B$ contains $4$ white,$2$ red and $3$ black balls. If two balls are drawn at random,without replacement,from a randomly selected box and one ball turns out to be white while the other ball turns out to be red,then the probability that both balls are drawn from box $B$ is

There are three bags $X$,$Y$,and $Z$. Bag $X$ contains $5$ one-rupee coins and $4$ five-rupee coins; Bag $Y$ contains $4$ one-rupee coins and $5$ five-rupee coins; and Bag $Z$ contains $3$ one-rupee coins and $6$ five-rupee coins. $A$ bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability that it came from bag $Y$ is: