Question

Assume that we define two events for the random experiment, which is rolling a fair die. In the first event the result of a roll is an odd number. In the second event the result of a roll is less than 3 . Show that these two events are independent.

Answer 1

To demonstrate the independence of two events when rolling a fair die, one must show that P(A and B) = P(A) × P(B). In this situation, the events of rolling an odd number and rolling a number less than 3 are shown to be independent, as their combined probability matches the product of their individual probabilities.

Step-by-step explanation:

Independence of Events in Probability

To show that two events are independent, one must demonstrate that the occurrence of one event does not affect the probability of the occurrence of the other event. In the case of rolling a fair die, the event of rolling an odd number (Event A) includes the outcomes {1, 3, 5}, and the event of rolling a number less than 3 (Event B) includes the outcomes {1, 2}.

The probability of Event A, P(A), is 3/6 or 1/2 since there are three odd numbers out of six possible outcomes in a standard die. The probability of Event B, P(B), is 2/6 or 1/3 since there are two outcomes less than 3 out of six possible outcomes. To test for independence, we check if P(A and B) = P(A) × P(B).

The only outcome that is both odd and less than 3 is '1'. Thus, P(A and B) is 1/6 since there's only one outcome that satisfies both events in the sample space of six. When we calculate P(A) × P(B), we get (1/2) × (1/3) = 1/6, which is equal to P(A and B). Therefore, the events A (rolling an odd number) and B (rolling a number less than 3) are independent events because P(A) × P(B) = P(A and B).