Question

Answer the following questions with TRUE or FALSE. It is good practice to explain your answers. (a) The intersection of two events A and B can be larger than the union of the same two events A and B. (b) The probability of a single event A must be smaller than or equal to the union of two events A and B. (c) The condition probability of A given B must be smaller than the intersection of the same two events A and B (d) If two events are independent, that P{A or B} = P{A and B} means

Answer 1

The answers to the statements are (a) False, (b) True, (c) False, and (d) True.

Step-by-step explanation:

(a) False. The intersection of events A and B represents the outcomes that are common to both events. The union of events A and B represents all the outcomes in either event A or event B. Therefore, the intersection cannot be larger than the union.

(b) True. The probability of a single event A is always between 0 and 1. When we consider the union of events A and B, the probability can increase because we are considering more outcomes.

(c) False. The conditional probability of A given B represents the probability of event A occurring, given that event B has already occurred. The intersection of events A and B represents the outcomes that are common to both events. The conditional probability can be greater than or equal to the intersection.

(d) True. If two events A and B are independent, then the probability of A or B is equal to the sum of their individual probabilities minus the probability of their intersection, which is equal to P(A) + P(B) - P(A and B). However, if P(A and B) = 0, then P(A or B) = P(A) + P(B), which means that P(A or B) = P(A and B).

Answer 2

The statement about intersection being larger than the union is false, the probability of a single event being smaller than the union is true, the claim about conditional probability being smaller than the intersection is false, and the statement regarding the equivalence of the probabilities of 'A OR B' and 'A AND B' for independent events is false.

Step-by-step explanation:

Let's evaluate each of the statements:

• (a) False. The intersection of two events A and B, denoted as P(A AND B), represents the probability of both events occurring together. The union of two events A and B, denoted as P(A OR B), includes all possibilities where either A, B, or both occur. It is logically impossible for the intersection to be larger than the union.
• (b) True. The probability of a single event A is always less than or equal to the union of the event A with any other event B, because the union includes all outcomes from A plus additional outcomes from B, if there are any.
• (c) False. The conditional probability P(A|B) is the probability of A given that B has occurred. This value is calculated as P(A AND B) / P(B), where P(B) > 0. By definition, this value must be less than or equal to 1, hence it cannot be larger than P(A AND B), which is a probability value itself.
• (d) False. If two events are independent, the probability that A or B occurs, P(A OR B), is equal to the sum of their individual probabilities minus the probability of their intersection, P(A) + P(B) - P(A AND B). If P(A OR B) were equal to P(A AND B), it would suggest that they never occur separately, which is not the definition of independent events.