exercise:A286091bf2: 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> (Feller<ref group="Notes" >W. Feller, ''Introduction to Probability Theory and Its Applications,'' 3rd ed., vol. 1 (New York: John Wiley and Sons, 1968), p. 240.</ref>) A large number, <math>N</math>, of people are subjected to a blood test. Th...") |
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(Feller<ref group="Notes" >W. Feller, ''Introduction to Probability Theory and Its Applications,'' 3rd ed., vol. 1 (New York: John Wiley and | |||
Probability Theory and Its Applications,'' 3rd ed., vol. 1 (New York: John Wiley and | |||
Sons, 1968), p. 240.</ref>) A large number, <math>N</math>, of people are subjected to a blood | Sons, 1968), p. 240.</ref>) A large number, <math>N</math>, of people are subjected to a blood | ||
test. This can be administered in two ways: (1) Each person can be tested separately, in | test. This can be administered in two ways: (1) Each person can be tested separately, in | ||
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that the probability <math>p</math> that a test is positive is the same for all people and that | that the probability <math>p</math> that a test is positive is the same for all people and that | ||
these events are independent. | these events are independent. | ||
<ul><li> Find the probability that the test for a pooled sample of <math>k</math> people will be | <ul style="list-style-type:lower-alpha"><li> Find the probability that the test for a pooled sample of <math>k</math> people will be | ||
positive. | positive. | ||
</li> | </li> | ||
Latest revision as of 18:15, 14 June 2024
(Feller[Notes 1]) A large number, [math]N[/math], of people are subjected to a blood test. This can be administered in two ways: (1) Each person can be tested separately, in this case [math]N[/math] test are required, (2) the blood samples of [math]k[/math] persons can be pooled and analyzed together. If this test is negative, this one test suffices for the [math]k[/math] people. If the test is positive, each of the [math]k[/math] persons must be tested separately, and in all, [math]k + 1[/math] tests are required for the [math]k[/math] people. Assume that the probability [math]p[/math] that a test is positive is the same for all people and that these events are independent.
- Find the probability that the test for a pooled sample of [math]k[/math] people will be positive.
- What is the expected value of the number [math]X[/math] of tests necessary under plan (2)? (Assume that [math]N[/math] is divisible by [math]k[/math].)
- For small [math]p[/math], show that the value of [math]k[/math] which will minimize the expected number of tests under the second plan is approximately [math]1/\sqrt p[/math].
Notes