In four schools $B_1, B_2, B_3, B_4$,the percentage of girl students is $12, 20, 13, 17$ respectively. From a school selected at random,one student is picked up at random and it is found that the student is a girl. The probability that the school selected is $B_2$ is:

$A$ manufacturing company of bulbs has $3$ units $A, B$ and $C$ which produce $25 \%$,$35 \%$ and $40 \%$ of the bulbs respectively. Out of the bulbs produced by $A, B, C$ units,$5 \%, 4 \%$ and $2 \%$ are defective respectively. If a bulb is chosen at random and found to be defective,then the probability that it is produced by unit $B$ is

In a test,a student either guesses,copies,or knows the answer to a multiple-choice question with four choices. The probability that he guesses is $1/3$ and the probability that he copies the answer is $1/6$. The probability that his answer is correct,given that he copied it,is $1/8$. The probability that he knew the answer to the question,given that he answered it correctly,is

Let $U_1$ and $U_2$ be two urns such that $U_1$ contains $3$ white and $2$ red balls,and $U_2$ contains only $1$ white ball. $A$ fair coin is tossed. If head appears,then $1$ ball is drawn at random from $U_1$ and put into $U_2$. However,if tail appears,then $2$ balls are drawn at random from $U_1$ and put into $U_2$. Now $1$ ball is drawn at random from $U_2$.
$1.$ The probability of the drawn ball from $U_2$ being white is
$(A)$ $\frac{13}{30}$ $(B)$ $\frac{23}{30}$ $(C)$ $\frac{19}{30}$ $(D)$ $\frac{11}{30}$
$2.$ Given that the drawn ball from $U_2$ is white,the probability that head appeared on the coin is
$(A)$ $\frac{17}{23}$ $(B)$ $\frac{11}{23}$ $(C)$ $\frac{15}{23}$ $(D)$ $\frac{12}{23}$
Give the answer for question $1$ and $2.$

$A$ 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

$A$ bag $P$ contains $5$ white marbles and $3$ black marbles. Four marbles are drawn at random from $P$ and are put in an empty bag $Q$. If a marble drawn at random from $Q$ is found to be black,then the probability that all the three black marbles in $P$ were transferred to the bag $Q$ is: