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

$A$ 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.

In a factory which manufactures bolts,machines $A, B$ and $C$ manufacture respectively $25\%, 35\%$ and $40\%$ of the bolts. Of their outputs,$5, 4$ and $2$ percent are respectively defective bolts. $A$ bolt is drawn at random from the product and is found to be defective. What is the probability that it is manufactured by the machine $B$?

$A$ 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

$A$ bag contains $8$ balls,whose colors are either white or black. $4$ balls are drawn at random without replacement and it was found that $2$ balls are white and $2$ balls are black. The probability that the bag originally contained an equal number of white and black balls is:

An urn contains $5$ balls. Two balls are drawn at random and they are found to be white. The probability that all the balls in the urn are white is: