Question 1192064
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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 {{{x[1] = 0.107835}}}


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 {{{x[2] = 0.107835}}}


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 {{{x[3] = 0.218435}}}


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 {{{x[4] = 0.405665}}}


Cases {{{x[1]}}} through {{{x[3]}}} represent situations where exactly two judges get the right decision.
Case {{{x[4]}}} is when all three judges make the correct ruling.
All four represent when at least two judges get the correct ruling.


Add up the {{{x[1]}}} through {{{x[4]}}}


{{{x[1]+x[2]+x[3]+x[4] = 0.107835+0.107835+0.218435+0.405665 = 0.83977}}}


The probability that at least two judges reach the correct decision, and therefore get the correct overall ruling, is 0.83977


Cases {{{x[1]}}}, {{{x[3]}}}, and {{{x[4]}}} represent situations where judge A made the correct ruling that led to the overall ruling being correct.
The sum of these x values is {{{x[1]+x[3]+x[4] = 0.731935}}}


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 {{{x[2]}}} 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? 
<font color=red>0.83977</font>


If their collective decision was correct, what is the chance of judge's A decision being the right one? 
<font color=red>0.871590</font>


What is the probability that only B and C had the right decision? 
<font color=red>0.107835</font>
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