exercise:Dd22e9bec9: Difference between revisions
From Stochiki
(Created page with "A group of 100 patients is tested, one patient at a time, for three risk factors for a certain disease until either all patients have been tested or a patient tests positive f...") |
mNo edit summary |
||
| Line 1: | Line 1: | ||
A group of 100 patients is tested, one patient at a time, for three risk factors for a certain disease until either all patients have been tested or a patient tests positive for more than one of these three risk factors. For each risk factor, a patient tests positive with probability <math>p</math>, where | A group of 100 patients is tested, one patient at a time, for three risk factors for a certain disease until either all patients have been tested or a patient tests positive for more than one of these three risk factors. For each risk factor, a patient tests positive with probability <math>p</math>, where <math>0 < p < 1</math>. The outcomes of the tests across all patients and all risk factors are mutually independent. | ||
Determine an expression for the probability that exactly <math>n</math> patients are tested, where <math>n</math> is a positive integer less than 100. | Determine an expression for the probability that exactly <math>n</math> patients are tested, where <math>n</math> is a positive integer less than 100. | ||
Latest revision as of 11:24, 4 May 2023
A group of 100 patients is tested, one patient at a time, for three risk factors for a certain disease until either all patients have been tested or a patient tests positive for more than one of these three risk factors. For each risk factor, a patient tests positive with probability [math]p[/math], where [math]0 \lt p \lt 1[/math]. The outcomes of the tests across all patients and all risk factors are mutually independent.
Determine an expression for the probability that exactly [math]n[/math] patients are tested, where [math]n[/math] is a positive integer less than 100.
- [math][1-3p^2(1-p)]^{n-1}[3p^2(1-p)][/math]
- [math][1-3p^2(1-p) - p^3]^{n-1}[3p^2(1-p) + p^3][/math]
- [math][1-3p^2(1-p) - p^3]^{n-1}[3p^2(1-p) + p^3]^{n-1}[/math]
- [math]n[1-3p^2(1-p) - p^3]^{n-1}[3p^2(1-p) + p^3][/math]
- [math]3[(1 − p )^{n −1} p][1-(1-p)^{n-1}p] + [(1-p)^{n-1}p]^3[/math]
Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.