Suppose I have 3 friends, each of them are independent and each of them have the probability of p to tell the truth about something (e.g. company, weather, stock, etc.).
Suppose something happened (e.g. company is holding a party other than not holding a party, weather is good other than bad, stock is rising other than drop, etc.), I want to calculate what is the probability 2 of them telling truth, and 1 of them telling wrong?
I think the answer is
But what is the probability of something is true, if 2 of the friends telling me truth, but 1 of friends telling me it is not truth?
P (true | T, T, F) = P (T,T,F|true) * P(true) / P(T,T,F) demominator P (T,T,F) = P(true) * P(T,T,F|true) + P(false) * P (T,T,F|false) Suppose prior P(true) = 1/2 and P(false) = 1-1/2 = 1/2, then it become P (T,T,F) = x*3*p*p*1/2 + 1/2*3*p*(1-p)*(1-p) numerator = x*3*p*p*(1-p)1/2 final result = p
Suppose I consult more friends, but still they have the same ratio telling the true or not, so for 6 friends, 4 of them telling truth and 2 of them telling false The result becomes,
P (true | T,T,T,T,F,F) = P (T,T,T,T,F,F|true) * P(true) / P(T,T,T,T,F,F) numerator = C(2,6)*1/2*p^4*(1-p)^2 demominator = C(2,6)*1/2*p^4*(1-p)^2 + C(2,6)*1/2*p^2*(1-p)^4 C(2,6) means select 2 from 6, which is 15 final result is p^2 / (p^2 + (1-p)^2)
If I want to compare if asking 3 friends has higher probability of truth, or asking 6 friends, I just need to compare between p and p^2 / (p^2 + (1-p)^2),
I did calcualation, which gets as long as (2p-1)*(p-1) > 0, asking more friends, is better, which result in p<1/2, it seems counter intuitive?
My intuitive is, asking more friends is better if any individual friend has high confidence (p>1/2)? Anything wrong in my calculation?