Question

Determine whether the event is independent or dependent.

A red ball is drawn from a bag containing three red balls and two black balls. It is replaced, and then a black ball is drawn from the bag.
a) Independent
b) Dependent

Answer 1

The draw of a red ball and then a black ball from a bag, with the red ball being replaced, constitutes two independent events because the replacement restores the original composition of the bag and does not affect the probability of the second draw.

Step-by-step explanation:

When determining whether an event is independent or dependent, one must consider the impact of one event on the occurrence of the second event. In this specific case, a red ball is drawn from a bag containing three red balls and two black balls and then replaced, followed by the drawing of a black ball. Since the red ball is replaced, the composition of the bag is the same for both draws, making each draw an independent event from the other.

Here is the reasoning:

• After the red ball is replaced, the probability of drawing a black ball from the bag remains the same as it was before the red ball was drawn.
• The formula P(A AND B) = P(A)P(B) can be applied because the replacement of the red ball ensures that the probability of event B (a black ball being drawn) is not affected by the occurrence of event A (a red ball being drawn).

Thus, the act of drawing a red ball and then drawing a black ball, when the red ball is replaced, are independent events.