exercise:42db0d031a: Difference between revisions
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(Created page with "<div class="d-none"><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></div> <ul><li> 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...") |
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Suppose that you are looking in your desk for a letter from some time ago. Your desk has eight | 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 | drawers, and you assess the probability that it is in any particular drawer is 10% (so there is | ||
a 20 | 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 | 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 in the first <math>i</math> drawers, where <math>0 \le i \le 7</math>. Let <math>p_i</math> denote the probability that | ||
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will be found in some subsequent drawer (both <math>p_i</math> and <math>q_i</math> are conditional probabilities, since | 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 | 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.<ref group="Notes" >R. | the <math>p_i</math>'s increase and the <math>q_i</math>'s decrease. (This problem is from Falk et al.<ref group="Notes" >R. | ||
Falk, A. | Falk, A. Lipson, and C. Konold, “The ups and downs of the hope function in a fruitless search,” | ||
in ''Subjective Probability,'' G.\ Wright and P.\ Ayton, (eds.) (Chichester: Wiley, 1994), pgs. | in ''Subjective Probability,'' G.\ Wright and P.\ Ayton, (eds.) (Chichester: Wiley, 1994), pgs. | ||
353-377.</ref>) | 353-377.</ref>) | ||
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numbers: Alarming disease data are frequently flawed,” ''Wall Street Journal,'' 11 April 1996, p. | numbers: Alarming disease data are frequently flawed,” ''Wall Street Journal,'' 11 April 1996, p. | ||
B1.</ref> For the ages 20, 30, 40, 50, and 60, the probability of a | B1.</ref> 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 | 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 | respectively. At the same set of ages, the probability of a woman in the U.S.\ eventually developing cancer is | ||
39.6 | 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? | explanation for these data? | ||
</li> | </li> | ||
Revision as of 00:21, 13 June 2024
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- 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.[Notes 1])
- The following data appeared in an article in the Wall Street Journal.[Notes 2] 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
- R. Falk, A. Lipson, and C. Konold, “The ups and downs of the hope function in a fruitless search,” in Subjective Probability, G.\ Wright and P.\ Ayton, (eds.) (Chichester: Wiley, 1994), pgs. 353-377.
- C. Crossen, “Fright by the numbers: Alarming disease data are frequently flawed,” Wall Street Journal, 11 April 1996, p. B1.