Mathematics — Probability · conditional-probability-search
Problem
(a) Suppose that you are looking in your desk for a letter from some time ago. Your desk has eight drawers, and you assess the probability that it is in any particular drawer is 10\% (so there is a 20\% chance that it is not in the desk at all). Suppose now that you start searching systematically through your desk, one drawer at a time. In addition, suppose that you have not found the letter in the first i drawers, where 0≤i≤7. Let pi denote the probability that the letter will be found in the next drawer, and let qi denote the probability that the letter will be found in some subsequent drawer (both pi and qi are conditional probabilities, since they are based upon the assumption that the letter is not in the first i drawers). Show that the pi's increase and the qi's decrease.
(b) The following data appeared in an article in the Wall Street Journal. For the ages 20, 30, 40, 50, and 60, the probability of a woman in the U.S. developing cancer in the next ten years is 0.5\%, 1.2\%, 3.2\%, 6.4\%, and 10.8\%, respectively. At the same set of ages, the probability of a woman in the U.S. eventually developing cancer is 39.6\%, 39.5\%, 39.1\%, 37.5\%, and 34.2\%, respectively. Do you think that the problem in part (a) gives an explanation for these data?