Question 1200770
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Three judges with a majority vote have to make a decision. 
The judges decide independently of each other. 
Based on previous experience, it is known that the probabilities of each judge 
making the correct decision are p1, p2, p3. 
(a) What is the probability that the final decision will be correct? 
(b) If the final decision was correct, what is the probability that the first judge made the right decision? 
([p1, p2, p3] = [0.83, 0.83, 0.87])
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<pre>
We do not know what cases the judges consider and what judgment/decisions they make.

We only know that the decisions can be Right or Wrong.

Therefore, based on given information, we only can make a table of probabilities 
for all possible situations.


So, I made this table: it is below.


        1     2     3   verdict decided  Include     (+)   Individual
                        by a majority    not include (-)   probabilities 
                                                           for each possible court decision
  --------------------------------------------------------------------------------------------

    1   R     R     R  --->  R             +                0.83*0.83*0.87    
    2   R     R     W  --->  R             +                0.83*0.83*(1-0.87)    

    3   W     W     R  --->  W             -                
    4   W     W     W  --->  W             -                
       
    5   R     W     R  --->  R             +                0.83*(1-0.83)*0.87
    6   R     W     W  --->  W             -

    7   W     R     R  --->  R             +                (1-0.83)*0.83*0.87
    8   W     R     W  --->  W             -


The leftmost column is for the numbers of lines.

The digits 1, 2 and 3 in the horizontal upper line represent the judges.
"R" represents right decision; "W" represents wrong decision.

The symbols in the table below "1", "2" and "3" symbolize the decisions (R for right, W for wrong).

The symbols in the column named "verdict decided by a majority"
represent the logical consequence of the decisions made in columns 1, 2, and 3.
It is how the court makes its final decision, based on individual decisions of the judges.  
The arrows ( ---> ) show the logical implications  ("verdict decided by a majority").


Notice that in the table I listed ALL LOGICALLY POSSIBLE situations.
There are NO other possible situations that would be consistent with the problem.


According to the problem, the question is about the probability 
of the final court's decision to be right. So, in column "include or not include"
I write "+" for right decisions to include     them into the final count
     or "-" for wrong decisions to NOT include them into the final count.


In the rightmost column, I calculated the probabilities for each 
possible RIGHT decision of the curt.


From this consideration, the final probability of the right verdict of the court is 

    0.83*0.83*0.87 + 0.83*0.83*(1-0.87) + 0.83*(1-0.83)*0.87 + (1-0.83)*0.83*0.87 = 0.934414 (precise value).   (*)   <U>ANSWER</U>


You may round it in any way you want.
</pre>

Thus part (a) is complete.


The post-solution note:  


    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Beeing familiar with this detailed explanation and its logic, and looking in the structure 
    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;of the formula (*), you may produce much shorter solution in your head in couple of lines.


-------------------


This my solution is written as a description of an ALGORITHM 
calculating the desired probability based on given input data.



///////////////////////



<pre>
Now for part (b), it asks about the conditional probability.


        It is the ratio of two values.


The denominator is the value of the part (a) above.


The numerator is that three addends of the formula (*)

        0.83*0.83*0.87 + 0.83*0.83*(1-0.87) + 0.83*(1-0.83)*0.87,

where the first judge was right.


So, the answer for (b) is


{{{(0.83*0.83*0.87 + 0.83*0.83*(1-0.87) + 0.83*(1-0.83)*0.87)/(0.83*0.83*0.87 + 0.83*0.83*(1-0.87) + 0.83*(1-0.83)*0.87 + (1-0.83)*0.83*0.87)}}} = 0.8686  (rounded).   <U>ANSWER</U>
</pre>

At this point, &nbsp;the problem is solved in full.