$A$ bag contains $10$ balls out of which $k$ are red and $(10-k)$ are black,where $0 \le k \le 10$. If three balls are drawn at random without replacement and all of them are found to be black,then the probability that the bag contains $1$ red and $9$ black balls is:

$A$ bag contains $6$ balls. If $4$ balls are drawn at a time and all of them are found to be red,then the probability that exactly $5$ of the balls in the bag are red is

$A$ laboratory blood test is $99 \%$ effective in detecting a certain disease when it is in fact,present. However,the test also yields a false positive result for $0.5 \%$ of the healthy persons tested (that is,if a healthy person is tested,then,with probability $0.005,$ the test will imply he has the disease). If $0.1 \%$ of the population actually has the disease,what is the probability that a person has the disease given that his test result is positive?

$A$ person is known to speak the truth in $3$ out of $4$ occasions. If he throws a die and reports that it is six,then the probability that it is actually six is

Of the students in a college,it is known that $60 \%$ reside in a hostel and $40 \%$ are day scholars (not residing in a hostel). Previous year results report that $30 \%$ of all students who reside in a hostel attain $A$ grade and $20 \%$ of day scholars attain $A$ grade in their annual examination. At the end of the year,one student is chosen at random from the college and they have an $A$ grade. What is the probability that the student is a hostler?

There are three bags $B_1, B_2$ and $B_3$. The bag $B_1$ contains $5$ red and $5$ green balls,$B_2$ contains $3$ red and $5$ green balls,and $B_3$ contains $5$ red and $3$ green balls. Bags $B_1, B_2$ and $B_3$ have probabilities $\frac{3}{10}, \frac{3}{10}$ and $\frac{4}{10}$ respectively of being chosen. $A$ bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?
$(1)$ Probability that the selected bag is $B_3$ and the chosen ball is green equals $\frac{3}{20}$
$(2)$ Probability that the chosen ball is green equals $\frac{39}{80}$
$(3)$ Probability that the chosen ball is green,given that the selected bag is $B_3$,equals $\frac{3}{8}$
$(4)$ Probability that the selected bag is $B_3$,given that the chosen ball is green,equals $\frac{4}{13}$