Two digits are selected at random from the digits $1$ through $9$. If their sum is even,then the probability that both are odd is

$A$ doctor assumes that a patient has one of three diseases $d_1, d_2,$ or $d_3$. Before any test,he assumes an equal probability for each disease. He carries out a test that will be positive with probability $0.7$ if the patient has disease $d_1$,$0.5$ if the patient has disease $d_2$,and $0.8$ if the patient has disease $d_3$. Given that the outcome of the test was positive,what is the probability that the patient has disease $d_2$?

$A$ man is known to speak the truth $4$ out of $5$ times. He throws a die and reports that it is a six. The probability that it was actually a six 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 $HIV$?

An urn contains $6$ white and $9$ black balls. Two successive draws of $4$ balls are made without replacement. The probability,that the first draw gives all white balls and the second draw gives all black balls,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: