If random variable $\mathrm{X} ~ \mathrm{B}\left(\mathrm{n} = 5, \mathrm{P} = \frac{1}{3}\right)$, then P(2 < X < 4) is equal to

• $\frac{80}{243}$

• $\frac{40}{243}$

• $\frac{40}{343}$

• $\frac{80}{343}$

At random variable X $~$ B(n, p), if values of mean and variance of X are 18 and 12 respectively, then total number of possible values of X are

• 54

• 55

• 12

• 18

Probability that a person will develop immunity after vaccinations is 0.8. If 8 people are given the vaccine, then probability that all develop immunity is =

• (0.2)⁸

• (0.8)⁸

• 1

• ${}^8\mathrm{C}_6{\left(0.2\right)}^6{\left(0.8\right)}^2$

For the following distribution function F(x) of a rv.X.

x	1	2	3	4	5	6
F(x)	0.2	0.37	0.48	0.62	0.85	1

$\mathrm{P}\left(3 < \mathrm{x} \le 5\right)$ =

• 0.48

• 0.37

• 0.27

• 1.47

Probability of solving a particular question by person A is 1/3 and probability ofsolving that question by person B is 2/5. what is the probability of solving that question by atleast one of them ?

• $\frac{3}{5}$

• $\frac{7}{9}$

• $\frac{2}{5}$

• $\frac{2}{3}$

The probabilities that Mr. A and Mr. B will die within a year are $\frac{1}{2} \mathrm{and} \frac{1}{3}$ respectively, then the probability that only one of them will be alive at the end of the year, is

• $\frac{5}{6}$

• $\frac{1}{2}$

• $\frac{2}{3}$

• None of the above

The probability that a man will live 10 more years, is $\frac{1}{4}$ and the probability that his wife  will live 10 more years, is $\frac{1}{3}$. Then, what is the probability that neither will be alive in 10 yr ?

• $\frac{1}{2}$

• $\frac{3}{7}$

• $\frac{2}{3}$

• $\frac{1}{4}$

The probability that in a year of the 22nd century chosen at random, there will be 53 Sunday, is

• $\frac{3}{28}$

• $\frac{2}{28}$

• $\frac{7}{28}$

• $\frac{5}{28}$

If P(A) = 65, P(B) = 80, then P(A $\cap$ B) lies in the interval

• [30, 80]

• [35, 75]

• [4, 70]

• [0.45, 0.65]

For any two independent events E₁ and E₂ in a space S, $\mathrm{P}\left[\left({\mathrm{E}}_1 \cup {\mathrm{E}}_2\right) \cap \left({\mathrm{E}}_1 \cap {\mathrm{E}}_2\right)\right]$ is equal to

• $\le \frac{1}{4}$

• $> \frac{1}{4}$

• $\ge \frac{1}{2}$

• $> \frac{1}{2}$