Partition of Sample Space

Q. If A and B are independent events, then write expression for P(exactly one of A, B occurs).

Q.

If E and F are events such that P(E) =, P(F) = and P(E and F) =, find:(i) P(E or F), (ii) P(not E and not F).

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

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. $\frac{17}{30}$
4. $\frac{8}{15}$

Q.

From eighty cards numbered $1$ to $80$, two cards are selected randomly. The probability that both the cards have the numbers divisible by $4$ is given by

1. $\frac{21}{316}$

2. $\frac{19}{316}$

3. $\frac{1}{4}$

4. None of these

Q. Two fair coins are tossed once. Then the number of elements in the sample space is

Q. The set which is a partition of the sample space we get while throwing a die is

1. {(1, 2), (2, 3), (3, 4)}
2. {(1, 2), (5, 6)}
3. {(1, 2), (1, 2, 3, 4, 5, 6)}
4. {(1, 2, 3, 4), (5, 6)}

Q. The sample space for the indicated experiment, where a coin is tossed three times is:

1. $\{HHH,TTT\}$
2. $\{HHH,THH,HTH,HHT\}$
3. $\{HHH,THH,HTH,HHT,TTT,HTT\}$
4. $\{HHH,THH,TTH,THT,HTH,TTT,HTT,HHT\}$

Q. For a random experiment, all possible outcomes are called

1. numerical space.
2. event space.
3. sample space.
4. both b and c.

Q.

Does increasing the size of a sample necessarily make the sample more representative of a population? Give an example to support your explanation.

Q. Sample space for experiment in which a dice is rolled is

1. $4$
2. $8$
3. $12$
4. None of these

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 A and B are two events in a sample space S such that $P\left(A\right)\ne 0$ and $P\left(B\right)\ne 0$, then $P(A\cap B)$

1. $P\left(A\right).P\left(B\right)$
2. $P\left(A\right).P\left(\frac{B}{A}\right)$
3. $P\left(A\right)$
4. $P\left(B\right)$

Q. Tossing a coin is an example of .........

1. Infinite discrete sample space
2. Finite sample space
3. Continuous sample space
4. None of these

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. $\frac{17}{30}$

Q. Sample space for experiment in which two coins are tossed is

1. $8$
2. $4$
3. $2$
4. None of these

Q. The experiment is to repeatedly toss a coin until first tail shows up. Identify the type of the sample space.

1. Finite sample space
2. Continuous sample space
3. Infinite discrete sample space
4. None of these

Q. There are 50 marbles of 3 colors: blue yellow and black The probability of picking up a blue marble is 3/10 and that of picking up a yellow marble is 1/2 The probability of picking up a black ball is

1. 1/4
2. 4/5
3. 1/5
4. 1/10

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. $\frac{17}{30}$

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

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. $\frac{17}{30}$

Q. In a game, a man wins a rupee for a six and loses a rupee for any other number when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a six. Find the expected value of the amount he win / loses.

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

1. { (1, 2), (2, 3), (3, 4), (4, 5), (5, 6) }
2. { (1), (2), (3), (4), (5), (6) }
3. {(1, 2), (1, 2, 3, 4, 5, 6), }
4. { (1, 2, 3, 4), (5, 6) }

Q. In a class $30$% students fail in English; $20$% students fail in Hindi and $10$% students fail in English and Hindi both. A student is chosen at random, then the probability that he will fail English if he has failed in Hindi is $\frac{k}{2}$.The value of $k$ is

Q.

Events A and B are such that . State whether A and B are independent?

Q. Show that the relative $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\left\{\right(a,b):|a-b|\text{is even}\}$, is an equivalence relation. Show that all the elements of $\{1,3,5\}$ are related to each other and all the elements of $\{2,4\}$ are related to each other. But no element of $\{1,3,5\}$ is related to any element of $\{2,4\}$.

Q.

Events A and B are such that . State whether A and B are independent?

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 $A$ and $B$ are two events, the probability that exactly one of them occurs is given by

1. $P\left(A\right)+P\left(B\right)-2P(A\cap B)$
2. $P(A\cap B^{\prime })+P(A^{\prime }\cap B)$
3. $P(A\cap B)-P(A\cap B)$
4. $P\left(A^{\prime }\right)+P\left(B^{\prime }\right)-2P(A^{\prime }\cap B^{\prime })$

Q. $R=\left\{\right(a,b):a=b\}$