[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

• 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 [math]i[/math] drawers, where [math]0 \le i \le 7[/math]. Let [math]p_i[/math] denote the probability that the letter will be found in the next drawer, and let [math]q_i[/math] denote the probability that the letter will be found in some subsequent drawer (both [math]p_i[/math] and [math]q_i[/math] are conditional probabilities, since they are based upon the assumption that the letter is not in the first [math]i[/math] drawers). Show that the [math]p_i[/math]'s increase and the [math]q_i[/math]'s decrease. (This problem is from Falk et al.)
• 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?

Notes