Assume that a box contains four red jelly beans and two green ones. We consider the event that a red bean is drawn. Now suppose I pick a jelly bean from the same box without looking, but I do not put the bean back in the box. Then I choose a bean again.

Question

Are the two events independent

Answer 1

No, the two events are not independent. The probability of drawing a red jelly bean on the first draw is 4/6, or 2/3. However, if a red bean is drawn on the first draw and is not replaced, then the probability of drawing a red jelly bean on the second draw becomes 3/5 instead of 4/6. This is because there are now only 3 red jelly beans left in the box, and the total number of jelly beans remaining in the box is only 5. Therefore, the probability of drawing a red bean on the second draw depends on the outcome of the first draw, and the events are not independent.