State the axioms of probability. Let and be two events associated with a sample space Define,...

 

Transcribed Image Text:

State the axioms of probability. Let and be two events associated with a sample space Define, in terms of probability, what it means to say and are: i) Exhaustive Independent (c) What is the difference between Mutually exclusive and Complementary events? (d) Show that if are mutually exclusive events, then 2. (a) Two events and associated with a sample space of an experiment are such that, and. Determine whether and are independent. (b) and and Two events and are such that, they are independent 3. 5. (b) (c) (b) (c) and. Find the value of and, where are the complements of events and respectively. What is a partition of a sample space? Differentiate between mutually independent and pairwise independent events. and are three mutually independent events, such that. 4. (a) Show that if and are any two independent events on the sample space of an experiment, then and are also independent, where and are the complements of and respectively. Show that if then. and. What is the conditional probability that even occurs given that at least one of and occur? For three events, and, we know that and are independent, and are independent, and are disjoint. 4 12 Find and exclusive. Suppose that i. For what values of are and mutually exclusive? For this value of, are A and B independent? ii. For what values of, are and independent? Determine whether for this value of x. A and B are mutually (a) Gideon rolls two (2) six-sided dice once. What is the probability that the sum of the (b) outcomes of both dice are odd or divisible by 5? Two (2) dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on two different numbers? 7. If the letters of the word PROBLEM are arranged at random without repetition, find the probability that there will be five letters between O and E. 9. (b) A committee of 5 is to be selected from a group of 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women? 8. (a) An urn contains white and black balls. If two balls are drawn at random, find the probability that the balls will be of different colours when and (m) same colour when and are positive numbers. (b) Thirty (30) identical balls except for colour are placed in three urns such that, Urn I contains 6 white, 5 red and 4 black balls. Urn II contains 2 white, 3 red and 5 black balls. Urn III contains only 5 black balls. If a ball is selected, what is the probability that it is black? Two numbers are chosen at random from among the numbers 1 to 10 without replacement. Find the probability that the second number chosen is 5. (b) A box contains 8 yellow, 5 green and 7 black balls identical to each other except for colour. 3 balls are drawn at random one after the other without replacement. Find the probability that: (i) there will be one ball of each colour. exactly one ball will be black (ii) () at least one ball will be black (iv) the second ball will be black () the second ball will be black given that the first is black. State the axioms of probability. Let and be two events associated with a sample space Define, in terms of probability, what it means to say and are: i) Exhaustive Independent (c) What is the difference between Mutually exclusive and Complementary events? (d) Show that if are mutually exclusive events, then 2. (a) Two events and associated with a sample space of an experiment are such that, and. Determine whether and are independent. (b) and and Two events and are such that, they are independent 3. 5. (b) (c) (b) (c) and. Find the value of and, where are the complements of events and respectively. What is a partition of a sample space? Differentiate between mutually independent and pairwise independent events. and are three mutually independent events, such that. 4. (a) Show that if and are any two independent events on the sample space of an experiment, then and are also independent, where and are the complements of and respectively. Show that if then. and. What is the conditional probability that even occurs given that at least one of and occur? For three events, and, we know that and are independent, and are independent, and are disjoint. 4 12 Find and exclusive. Suppose that i. For what values of are and mutually exclusive? For this value of, are A and B independent? ii. For what values of, are and independent? Determine whether for this value of x. A and B are mutually (a) Gideon rolls two (2) six-sided dice once. What is the probability that the sum of the (b) outcomes of both dice are odd or divisible by 5? Two (2) dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on two different numbers? 7. If the letters of the word PROBLEM are arranged at random without repetition, find the probability that there will be five letters between O and E. 9. (b) A committee of 5 is to be selected from a group of 6 men and 9 women. If the selection is made randomly, what is the probability that the committee consists of 3 men and 2 women? 8. (a) An urn contains white and black balls. If two balls are drawn at random, find the probability that the balls will be of different colours when and (m) same colour when and are positive numbers. (b) Thirty (30) identical balls except for colour are placed in three urns such that, Urn I contains 6 white, 5 red and 4 black balls. Urn II contains 2 white, 3 red and 5 black balls. Urn III contains only 5 black balls. If a ball is selected, what is the probability that it is black? Two numbers are chosen at random from among the numbers 1 to 10 without replacement. Find the probability that the second number chosen is 5. (b) A box contains 8 yellow, 5 green and 7 black balls identical to each other except for colour. 3 balls are drawn at random one after the other without replacement. Find the probability that: (i) there will be one ball of each colour. exactly one ball will be black (ii) () at least one ball will be black (iv) the second ball will be black () the second ball will be black given that the first is black.