$A$ card from a pack of $52$ cards is lost. From the remaining cards of the pack,two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond.

There are two balls in an urn. Each ball can be either white or black. If a white ball is put into the urn and thereafter a ball is drawn at random from the urn,then the probability that it is white is

$A$ doctor assumes that a patient has one of three diseases $d_1, d_2,$ or $d_3$. Before any test,he assumes an equal probability for each disease. He carries out a test that will be positive with probability $0.7$ if the patient has disease $d_1$,$0.5$ if the patient has disease $d_2$,and $0.8$ if the patient has disease $d_3$. Given that the outcome of the test was positive,what is the probability that the patient has disease $d_2$?

$A$ man is known to speak the truth $7$ out of $10$ times. After throwing a die with $100$ faces marked $1, 2, 3, \dots, 100$,the man reports that he got a prime number. What is the probability that it is actually a prime number?

Bag $B_1$ contains $6$ white and $4$ blue balls,Bag $B_2$ contains $4$ white and $6$ blue balls,and Bag $B_3$ contains $5$ white and $5$ blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white,then the probability that the ball is drawn from Bag $B_2$ is:

$A$ man speaks truth $2$ out of $3$ times. He picks one of the natural numbers in the set $S=\{1, 2, 3, 4, 5, 6, 7\}$ and reports that it is even. The probability that it is actually even is