There are $10$ coins in a box, out of which $8$ are normal and the remaining are with heads on both sides. $A$ coin is chosen at random from the box and tossed $6$ times. If it shows heads each time, then the probability that the selected coin has heads on both sides is

$A$ 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

$A$ 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

Urn $A$ contains $6$ red and $4$ black balls and urn $B$ contains $4$ red and $6$ black balls. One ball is drawn at random from urn $A$ and placed in urn $B$. Then one ball is drawn at random from urn $B$ and placed in urn $A$. If one ball is now drawn at random from urn $A$,the probability that it is found to be red,is

$A$ dealer gets refrigerators from $3$ different manufacturing companies $C_1, C_2$ and $C_3$. $25 \%$ of his stock is from $C_1, 35 \%$ from $C_2$ and $40 \%$ from $C_3$. The percentages of receiving defective refrigerators from $C_1, C_2$ and $C_3$ are $3 \%, 2 \%$ and $1 \%$ respectively. If a refrigerator sold at random is found to be defective by a customer, then the probability that it is from $C_2$ is

An examination is attempted by $5000$ graduates, $2000$ post-graduates, and $1000$ doctorate holders. The probabilities that a graduate, a post-graduate, and a doctorate holder will pass the examination are $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{4}{5}$ respectively. If one of the examinees passed the examination, then the probability that he is a post-graduate is: