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

$A$ bag contains $6$ balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least $5$ black balls 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?

Bag $I$ contains $3$ red and $4$ black balls. Bag $II$ contains $5$ red and $6$ black balls. If one ball is drawn at random from one of the bags, and it is found to be red, then the probability that it was drawn from Bag $II$, is

$A$ disease affects two-thirds of the population of a country. $A$ test for the disease gives the correct outcome with probability $\frac{2}{3}$. $A$ person $X$ tests positive for the disease. The probability that $X$ has the disease is

In a certain city, the ratio of men to women is $5:4$. It is found that $80\%$ of men and $90\%$ of women are literate. If a person selected at random is found to be illiterate, then the probability that the person is a man is....