Beginning statistics students are often puzzled by two characteristics of distributions in this chapter: (1) The trials are independent, and (2) the probability of a success remains constant from trial to trial. Students often think these two characteristics are the same. The questions in this exercise point out the difference. Consider a hypergeometric distribution where N = 3, X = 2, and n = 2.
a. Mathematically demonstrate that the trials for this experiment are dependent by calculating the probability of obtaining a success on the second trial if the first trial resulted in a success. Repeat this calculation if the first trial was a failure. Use these two probabilities to prove that the trials are dependent.
b. Now calculate the probability that a success is obtained on each of the three respective trials and, therefore, demonstrate that the trials are dependent but that the probability of a success is constant from trial to trial.