Question

4. Suppose a person enters a doctor's office and has four equal outcomes: healthy and return home (1), sick but return to work (2), sick and require diagnostic testing (3), sick and admit to hospital (4). Calculate the probabilities for the following scenarios: a) If one people enter the doctor's office, how likely will the person be determined sick and need to be admitted to the hospital? b) If two people enter the doctor's office, how likely will both of them be determined sick and need to be admitted to the hospital? We know the two people's health condition are independent. c) If 5 people enter the doctor's office, how likely will more than 3 be deemed sick but can return to work? We know the two people's health condition are independent (extra credit).

Answer 1

$4.~(a)~Since~the~4~outcomes~are~equally~likely~then\\ P(healthy~and~return~home)=P(sick~but~return~to~work)\\ =P(sick~and~require~diagnostic~testing)=P(sick~and~admit~to~hospital)\\ =\frac{1}{4}.\\ Then~the~probability~that~the~person~be~determined~sick~and~need~to~be~admitted\\ to~the~hospital=\frac{1}{4}=0.25.\\ (b)~Since~the~two~people's~health~condition~are~independent~so\\ required~probability=(1/4)^2=0.0625\\ (c)~Let~X~be~a~random~variable~denoting~no.~of~persons~deemed~sick~but~can~return~to~work.\\ Then~X\sim Binomial(5,1/4).\\ P(X>3)=P(X=4)+P(X=5)=\binom{5}{4}(1/4)^4(3/4)+\binom{5}{5}(1/4)^5=0.0156$