A bag contains five red balls and four blue balls. I take one at random, note its color, replace it, and shake the bag. I then take another ball at random, note its color, and replace it. The figure below shows the probability tree associated with this problem. Are the events of getting a blue ball on the first draw and getting a red ball on the second draw independent?
We recall that two events are independent if the incidence of one event does not affect the probability of the other event. We also know that the probability of the intersection of events 𝐴 and 𝐵 is equal to the probability of 𝐴 multiplied by the probability of 𝐵. The two events in this question are getting a blue ball on the first draw and getting a red ball on the second draw. We can see from the tree diagram that the probability of selecting a blue ball on the first draw is four-ninths. The probability of selecting a red ball on the second draw is five-ninths.
This is true irrespective of whether a red ball or blue ball is selected first. We know this is true as the first ball is being replaced. On each draw, there will be nine balls in the bag, five of which are red and four of which are blue. We can therefore conclude that the correct answer is yes. The events of getting a blue ball on the first draw and getting a red ball on the second draw are independent.