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 $HIV$?

Recent studies suggest that $12\%$ of the world population is left-handed. Depending on parents' hand usage, the chances of having left-handed children are as follows: $A$. Both parents are left-handed, chances of having left-handed children = $24\%$. $B$. Both parents are right-handed, chance of having left-handed children = $9\%$. $C$. Father left-handed and mother right-handed, chances of having left-handed children = $17\%$. $D$. Father right-handed and mother left-handed, chances of having left-handed children = $22\%$. Given $P(A) = P(B) = P(C) = P(D) = 1/4$ and $L$ denotes the event that the child is left-handed. What is the probability $P(A|L)$?

$A$ computer producing factory has only two plants $T_1$ and $T_2$. Plant $T_1$ produces $20 \%$ and plant $T_2$ produces $80 \%$ of the total computers produced. $7 \%$ of computers produced in the factory turn out to be defective. It is known that $P(\text{defective} | T_1) = 10 P(\text{defective} | T_2)$. $A$ computer produced in the factory is randomly selected and it is not defective. Then the probability that it is produced in plant $T_2$ is

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

$A$ 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.

$A$ boy speaks the truth in $3$ out of $5$ times. If he throws a die and reports that the number appeared on it is $5$, then the probability that it is actually $5$ is: