Probability Tes...

Let A and E be any two events with positive probabilities :
Statement - 1 : $$P \left (\displaystyle \frac{E}{A} \right) \geq P \left (\displaystyle \frac{A}{E} \right ) P(E)$$
Statement - 2 : $$P \left (\displaystyle \frac{A}{E}\right ) \geq P(A\cap E)$$

One ticket is selected at random from $$50$$ tickets numbered $$00,\ 01,\ 02, ... , 49$$. Then the probability that the sum of the digits on the selected ticket is $$8$$, given that the product of these digits is zero, equals

Three numbers are chosen at random without replacement from $$\{1, 2, 3, ..., 8\}$$. The probability that their minimum is $$3$$, given that their maximum is $$6$$, is:

For three events $$A, B$$ and $$C$$, $$P$$ (Exactly one of $$A$$ or $$B$$ occurs) $$= P$$ (Exactly one of $$B$$ or $$C$$ occurs) $$=P$$(Exactly one of $$C$$ or $$A$$ occurs) $$=\displaystyle\frac{1}{4}$$and $$P$$ (All the three events occur simultaneously)$$=\displaystyle\frac{1}{16}$$. Then the probability that at least one of the events occurs, is.

If $$\mathrm{C}$$ and $$\mathrm{D}$$ are two events such that $$\mathrm{C}\subset \mathrm{D}$$ and $$\mathrm{P}(\mathrm{D})\neq 0$$, then the correct statement among the following is 

Let A, B and C be three events, which are pair-wise independence and $$\overline{E}$$ denotes the complement of an event E. If $$P(A\cap B\cap C)=0$$ and $$P(C) > 0$$, then $$P[(\overline{A}\cap \overline{B})|C]$$ is equal to.

If $$A$$ and $$B$$ are any two events such that $$P(A) = \dfrac {2}{5}$$ and $$P(A\cap B) = \dfrac {3}{20}$$, then the conditional probability, $$P(A|(A'\cup B'))$$, where A' denotes the complement of $$A$$, is equal to:

It is given that the events A and $$B$$ are such that $$P(A)=\displaystyle \frac{1}{4},\ P(A|B)=\displaystyle \frac{1}{2}$$ and $$P(B|A)=\displaystyle \frac{2}{3}$$. Then $$P(B)$$ is 

A bag contains $$4$$ red and $$6$$ black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is

A box $$'A'$$ contains $$2$$ white, $$3$$ red and $$2$$ black balls. Another box $$'B'$$ contains $$4$$ white, $$2$$ red and $$3$$ black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box $$'B'$$ is