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:

There are $10$ coins in a box, out of which $8$ are normal and the remaining are with heads on both sides. $A$ coin is chosen at random from the box and tossed $6$ times. If it shows heads each time, then the probability that the selected coin has heads on both sides is

The probability that $A$ speaks the truth is $\frac{4}{5}$. $A$ coin is tossed. $A$ reports that a head appears. The probability that there was actually a head is

There are two boxes, each containing $10$ balls. In each box, some are black and the rest are white. $A$ ball is drawn at random from one of the boxes and it is found to be black. If the probability that the black ball drawn is from the second box is $\frac{1}{5}$, then the number of black balls in the first box is:

Suppose that the reliability of an $HIV$ test is specified as follows: Of people having $HIV$,$90\%$ of the tests detect the disease,but $10\%$ go undetected. Of people free of $HIV$,$99\%$ of the tests are judged $HIV$ negative,but $1\%$ are diagnosed as showing $HIV$ positive. From a large population of which only $0.1\%$ have $HIV$,one person is selected at random,given the $HIV$ test,and the pathologist reports him/her as $HIV$ positive. What is the probability that the person actually has $HIV$?

Three urns $A$,$B$,and $C$ contain $7$ red,$5$ black; $5$ red,$7$ black; and $6$ red,$6$ black balls,respectively. One of the urns is selected at random and a ball is drawn from it. If the ball drawn is black,then the probability that it is drawn from urn $A$ is: