A six-faced unbiased die is thrown twice and the sum of the numbers appearing on the upper face is observed to be 7. The probability that the number 3 has appeared atleast once, is

• $\frac{1}{5}$

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

If A, B and C are mutually exclusive and exhaustive events of a random experiment such that P(B) = $\frac{3}{2}$P(A) and P(C) = $\frac{1}{2}$P(B), then P(A ∪ C) equals to

• $\frac{10}{13}$

• $\frac{3}{13}$

• $\frac{6}{13}$

• $\frac{7}{13}$

Two persons A and B are throwing an unbiased six faced dice alternatively, with the condition that the person who throws 3 first wins the game. If A starts the game, then probabilities of A and B to win the same are, respectively

• $\frac{6}{11}, \frac{5}{11}$

• $\frac{5}{11}, \frac{6}{11}$

• $\frac{8}{11}, \frac{3}{11}$

• $\frac{3}{11}, \frac{8}{11}$

The letters of the word 'QUESTION' are arranged in a row at random. The probability that there are exactly two letters between Q and S is

• $\frac{1}{14}$

• $\frac{5}{7}$

• $\frac{1}{7}$

• $\frac{5}{28}$

$\mathrm{If} \frac{1 + 3\mathrm{P}}{3}, \frac{1 - 2\mathrm{P}}{2} \mathrm{are} \mathrm{probabilities} \mathrm{of} \mathrm{two} \mathrm{mutually} \mathrm{exclusive} \mathrm{events}, \mathrm{then} \mathrm{P} \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

• $\left[- \frac{1}{3}, \frac{1}{2}\right]$

• $\left(\frac{- 1}{2}, \frac{1}{2}\right)$

• $\left[- \frac{1}{3}, \frac{2}{3}\right]$

• $\left(\frac{- 1}{3}, \frac{2}{3}\right)$

A speaks truth in 75% of the cases and B in 80% of the cases. Then, the probability that their statements about an incident do not match, is

• $\frac{7}{20}$

• $\frac{3}{20}$

• $\frac{2}{7}$

• $\frac{5}{7}$

107.

$\mathrm{Given} \mathrm{below} \mathrm{is} \mathrm{the} \mathrm{distribution} \mathrm{of} \mathrm{a} \mathrm{random} \mathrm{variable} \mathrm{X}$

X = x	1	2	3	4
P(X = x)	$\mathrm{\lambda }$	$2\mathrm{\lambda }$	3$\mathrm{\lambda }$	4$\mathrm{\lambda }$

If $\mathrm{\alpha } = \mathrm{P}\left(\mathrm{X} < 3\right) \mathrm{and} \mathrm{\beta } = \mathrm{P}\left(\mathrm{X} > 2\right), \mathrm{then} \mathrm{\alpha } : \mathrm{\beta } = ?$

• 2 5

• 3 4

• 4 5

• 3 7

A bag P contains 5 white marbles and 3 black marbles. Four marbles are drawn at random from P and are put in an empty bag Q. If a marble drawn at random from Q is found to be black then the probability that all the three black marbles in P are transfered to the bag Q

• $\frac{1}{7}$

• $\frac{6}{7}$

• $\frac{1}{8}$

• $\frac{7}{8}$

${\mathrm{sech}}^{ - 1}\left(\frac{1}{\sqrt{2}}\right) + {\mathrm{csch}}^{ - 1}\left( - 1\right) = ?$

• 0

• $\sqrt{2} + 1$

• $\sqrt{2}$

• $\sqrt{2} - 1$

There are 10 intermediate stations on a railway line between two particular stations. The number ofways that a train can be made to stop at 3 of these intermediate stations so that no two of these halting stations are consecutive is

• 56

• 20

• 126

• 120