In an entrance test, there are multiple-choice questions. Each question has four possible answers, only one of which is correct. The probability that a student knows the answer to a question is $90\%$. If the student gets the correct answer to a question, what is the probability that they were guessing?

$A$ man is known to speak the truth $3$ out of $4$ times. He throws a die and reports that it is $6$. Then the probability that it is actually $6$ 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 $2.$

$A$ car manufacturing factory has two plants $X$ and $Y$. Plant $X$ manufactures $70 \%$ of cars and plant $Y$ manufactures $30 \%$ of cars. $80 \%$ of cars at plant $X$ and $90 \%$ of cars at plant $Y$ are rated as standard quality. $A$ car is chosen at random and is found to be standard quality. The probability that it has come from plant $X$ is

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

$A$ bag contains $4$ red and $4$ black balls,another bag contains $2$ red and $6$ black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.