Question 1192064: Three judges A, B and C must make a decision by majority's vote. They make their individual decision independently. It is known that the judges make a correct decision with probabilities A - 0.79, B - 0.65 and C 0.79. What is the chance that their decision will be correct? If their collective decision was correct, what is the chance of judge's A decision being the right one? Or what is the probability that only B and C had the right decision?
Answer by math_tutor2020(3817)   (Show Source):
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P(A) = probability that judge A makes the correct decision
P(B) and P(C) represent similar ideas for the other two judges.
Given probabilities
P(A) = 0.79
P(B) = 0.65
P(C) = 0.79
Their complements
P(~A) = 1 - P(A) = 1 - 0.79 = 0.21
P(~B) = 1 - P(B) = 1 - 0.65 = 0.35
P(~C) = 1 - P(C) = 1 - 0.79 = 0.21
which represent the probabilities of making the incorrect decision.
P(A and B only) = P(A, B, ~C)
P(A and B only) = P(A)*P(B)*P(~C)
P(A and B only) = 0.79*0.65*0.21
P(A and B only) = 0.107835
Let 
P(B and C only) = P(~A, B, C)
P(B and C only) = P(~A)*P(B)*P(C)
P(B and C only) = 0.21*0.65*0.79
P(B and C only) = 0.107835
Let 
P(A and C only) = P(A, ~B, C)
P(A and C only) = P(A)*P(~B)*P(C)
P(A and C only) = 0.79*0.35*0.79
P(A and C only) = 0.218435
Let 
P(A and B and C) = P(A)*P(B)*P(C)
P(A and B and C) = 0.79*0.65*0.79
P(A and B and C) = 0.405665
Let 
Cases  through  represent situations where exactly two judges get the right decision.
Case  is when all three judges make the correct ruling.
All four represent when at least two judges get the correct ruling.
Add up the through
The probability that at least two judges reach the correct decision, and therefore get the correct overall ruling, is 0.83977
Cases , , and represent situations where judge A made the correct ruling that led to the overall ruling being correct.
The sum of these x values is 
Dividing that second sum over the first sum calculated earlier will get us
0.731935/0.83977 = 0.871590
which is approximate.
This is the probability of judge A being correct given the overall ruling was correct.
This takes care of the second question mentioned.
For the third question, we go back to case  which is when judges B and C are correct, but judge A is not correct.
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Summary:
What is the chance that their decision will be correct?
0.83977
If their collective decision was correct, what is the chance of judge's A decision being the right one?
0.871590
What is the probability that only B and C had the right decision?
0.107835
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