Partition of Sample Space

Q.

Describe the sample space for the indicated experiment.
One die of red colour, one of white colour and one of blue colour are placed in a bag. One die is selected at random and rolled, its colour and the number on its upper most face is noted. Describe the sample space.

Q. A man is known to speak truth is 75% cases. If he throws an unbiased die and tells his friend that it is a six, then the probability that it is actually a six, is

1. $\frac{1}{6}$
2. $\frac{1}{8}$
3. $\frac{3}{4}$
   
4. $\frac{3}{8}$

Q.

A coin is tossed. If it shows a tail, we draw a ball from a box which contains 2 red and 3 black balls; if it shows a head, we throw a die. Find the sample space for this experiment.

Q. Which of the following set is partition of the sample space we get while throwing a die?

1. $\left\{\right(1,2),(2,3),(3,4),(4,5),(5,6\left)\right\}$
2. $\left\{\right(1),(2),(3),(4),(5),(6\left)\right\}$
3. $\left\{\right(1,2),(3,4\left)\right\}$
4. $\left\{\right(1,2,3,4),(5,6\left)\right\}$

Q. If events $E_1,E_2,...,E_n$ represent a partition of sample space S.
Then which of the following is/are correct?

1. $E_1\cap E_2=\varphi$
2. $E_1\cap E_2\ne \varphi$
3. $E_1\cup E_2\cup ...\cup E_n=S$
4. $P\left(E_i\right)>0\mathrm{for}\mathrm{all}i=1,2,\cdots ,n$

Q. If events $E_1,E_2,...,E_n$ represent a partition of sample space S.
Then which of the following is/are correct?

1. $E_1\cap E_2=\varphi$
2. $E_1\cap E_2\ne \varphi$
3. $E_1\cup E_2\cup ...\cup E_n=S$
4. $P\left(E_i\right)>0\mathrm{for}\mathrm{all}i=1,2,\cdots ,n$

Q. Sixteen players $P_1.P_2.\dots \dots \dots P_{16}$ play in a tournament. They are divided into eight pairs at random.From each pair a winner is decided on the basis of a game played between the two players of the pair.Assuming that all the players are of equal strength, the probability that exactly one of the two players $P_1$ and $P_2$ is among the eight winners is

1. $\frac{4}{15}$
2. $\frac{7}{15}$
3. $\frac{8}{15}$
4. $\frac{17}{30}$