A survey is carried out to find out the percentage of person

A survey is carried out to find out the percentage of persons affected by a certain disease. Because
some people may be inclined to deny being affected by this disease, each person questioned is given a six-faced die
and asked to roll it in secret and follow the following rules: if the shown number is 1, the interviewee must answer
NO, if the number shown is 6, the interviewee must answer YES and if the result is any other number then the
interviewee must say the truth (YES or NO). Since the number shown by the die is not revealed, it is impossible
to tell from the given answer if a particular interviewee is affected by the disease or not.
1. Suppose the probability of answering YES in the survey is found to be 2/3, what is the probability of
being affected by the disease?
2. Suppose the probability of being affected by the disease is 3/4. How likely is it that a person who
answers NO in affected by the disease? And a person who answers YES
3. The survey is carried out on 500 persons, 45 of which answer YES. Provide an estimate of the percent-
age of persons affected by the disease among the 500 interviewees.

Solution

Prob of answering yes = Prob (die shows 6 or die shows 2,3,4,5 and he has disease)

= +p =

where p is the probability of a person having the disease

p = 0.75=

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2) p = , Prob of saying no is affected by the disease = Prob (getting 1 and he is affected by the disease)

= =

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Prob of saying yes and also affected by disease = P(getting 6)+P(getting2,3,4,5 and having disease) =

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3) 45 out of 500 answered yes

Prob of having disease =p

Then getting 2,3,4,5 and having disease is 0.67p

Prob of having disease and answering yes = P(getting 2,3,4,5 and having disease) or P(getting 6)

p = prob of person affected by the disease =