An urn contains $5$ balls. Two balls are drawn at random and they are found to be white. The probability that all the balls in the urn are white is:

If a machine is correctly set up,it produces $90\%$ acceptable items. If it is incorrectly set up,it produces only $40\%$ acceptable items. Past experience shows that $80\%$ of the set ups are correctly done. If after a certain set up,the machine produces $2$ acceptable items,find the probability that the machine is correctly set up.

The probability that $A$ speaks the truth is $\frac{4}{5}$. $A$ coin is tossed. $A$ reports that a head appears. The probability that there was actually a head is

Bag $A$ contains $2$ white and $3$ red balls,and bag $B$ contains $4$ white and $5$ red balls. One ball is drawn at random from one of the two bags and it is found to be red. Find the probability that the ball was drawn from bag $B$.

One card is missing in a pack of $52$ playing cards. If two cards are drawn randomly from the remaining cards at a time and are found to be spades,then the probability that the missing card is not a spade is

There are three bags $X$,$Y$,and $Z$. Bag $X$ contains $5$ one-rupee coins and $4$ five-rupee coins; Bag $Y$ contains $4$ one-rupee coins and $5$ five-rupee coins; and Bag $Z$ contains $3$ one-rupee coins and $6$ five-rupee coins. $A$ bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability that it came from bag $Y$ is: