Question

An experiment consists of rolling a standard six-sided die once. Event A is rolling a 5, and event B is rolling an odd number. Are the events dependent or independent? Why? Select the option that correctly answers both questions.

a) Events A and B are dependent because P(A) ≠ P(B|A)
b) Events A and B are independent because P(A) = P(B|A) = 1/12
c) Events A and B are dependent because P(B) ≠ P(B|A)
d) Events A and B are independent because P(B) = P(B|A) = 1/2

Answer 1

Events A (rolling a 5) and B (rolling an odd number) on a six-sided die are independent because the occurrence of A does not affect the probability of B. Being independent, P(A) and P(B|A) are equal, hence the probability of event B given event A is the same as the probability of event B alone.

Step-by-step explanation:

The independence of two events in an experiment is investigated by checking whether the occurrence of one event affects the probability of the other. In the case of rolling a standard six-sided die, event A is rolling a 5, and event B is rolling an odd number. Event A has a probability of 1/6 as there is one '5' on the die. There are three odd numbers on a die (1, 3, and 5), so event B has a probability of 3/6 or 1/2. The probability of rolling a 5, given that an odd number is rolled (P(A|B)), remains at 1/6 because out of the three odd numbers (1, 3, and 5), only one is a 5.

Therefore, P(A) = 1/6 and P(B|A) = P(A) since rolling a 5 already satisfies the condition of rolling an odd number. This implies that the occurrence of A does not affect the probability of B, making the events independent. Thus, the correct answer is 'Events A and B are independent because P(B) = P(B|A) = 1/2', which is answer option d).