Let A and B be two mutually exclusive events with

$\mathrm{P}\left(\mathrm{A}\right) = \frac{1}{3} \mathrm{and} \mathrm{P}\left(\mathrm{B}\right) = \frac{1}{4}$. What is the value of $\mathrm{P}\left(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}\right) ?$

• $\frac{1}{6}$

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{5}{12}$

A committee of two persons is selected from two men and two women. The probability that the committee will have exactly one woman is :

• $\frac{1}{6}$

• $\frac{2}{3}$

• $\frac{1}{3}$

• $\frac{1}{2}$

Let a die be loaded in such a way that even faces are twice likely to occur as the odd faces. What is the probability that a prime number will show up when the die is tossed ?

• $\frac{1}{3}$

• $\frac{2}{3}$

• $\frac{4}{9}$

• $\frac{5}{9}$

Let the sample space consist of non-negative integers up to 50, X denote the numbers which are multiples of 3 and Y denote the odd numbers. Which of the following is/are correct ?

$$\begin{gathered} 1. \mathrm{P}\left(\mathrm{X}\right) = \frac{8}{25} \\ 2. \mathrm{P}\left(\mathrm{Y}\right) = \frac{1}{2} \end{gathered}$$

Select the correct answer using the code given below :

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2

For two events A and B, let PA) = $\frac{1}{2}, \mathrm{P}\left(\mathrm{A} \cup \mathrm{B}\right) = \frac{2}{3}$

$\mathrm{and} \mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = \frac{1}{6}. \mathrm{What} \mathrm{is} \mathrm{P}\left(\overline{\mathrm{A}} \cap \mathrm{B}\right) = ?$

• $\frac{1}{6}$

• $\frac{1}{4}$

• $\frac{1}{3}$

• $\frac{1}{2}$

Let A and B be two events with P(A) = $\frac{1}{3}$, $\mathrm{P}\left(\mathrm{B}\right) = \frac{1}{6}$ and $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right)$ = $\frac{1}{12}$. What is $\mathrm{P}\left(\mathrm{B}\overline{)\overline{\mathrm{A}}}\right)$ = ?

• $\frac{1}{5}$

• $\frac{1}{7}$

• $\frac{1}{8}$

• $\frac{1}{10}$

In a binomial distribution, the mean is $\frac{2}{3}$ and the variance is $\frac{5}{9}$. What is the probability that X = 2 ?

• $\frac{5}{36}$

• $\frac{25}{36}$

• $\frac{25}{216}$

• $\frac{25}{54}$

The probability that a ship safely reaches a port is $\frac{1}{3}$. The probability that out of 5 ships, at least 4 ships would arrive safely is

• $\frac{1}{243}$

• $\frac{10}{243}$

• $\frac{11}{243}$

• $\frac{13}{243}$

What is the probability that at least two persons out of a group of three persons were born in the same month (disregard year) ?

• $\frac{33}{144}$

• $\frac{17}{72}$

• $\frac{1}{144}$

• $\frac{2}{9}$

If $P\left(B\right) = \frac{3}{4}, \mathrm{P}\left(\mathrm{A} \cap \mathrm{B} \cap \overline{\mathrm{C}}\right) = \frac{1}{3} and \mathrm{P}\left(\overline{\mathrm{A}} \cap \mathrm{B} \cap \overline{\mathrm{C}}\right) = \frac{1}{3}$, then what is $\mathrm{P}\left(\mathrm{B} \cap \mathrm{C}\right) = ?$

• $\frac{1}{12}$

• $\frac{3}{4}$

• $\frac{1}{15}$

• $\frac{1}{9}$