Question 1200745
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A court consists of 3 judges. 
Two of them, independently of each other, make correct decisions with a probability of p. 
The third judge agrees with the first two decisions when they match. 
In case of different opinions of the first two judges, the third judge decides on his own 
and makes a mistake with a probability of q. 
What is the probability that the court will not make a mistake 
if the verdict is decided by a majority vote? ([p, q] = [0.5, 0.43])
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        Here I provide another solution for this given problem.

        It has a different form from the solution by the other tutor,  but produces the same answer.

        I wrote it simply to represent another way to organize and express the thoughts.



<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             +                0.5*0.5*1    
    2   W     W     W  --->  W             -                

    3   R     W     R  --->  R             +                0.5*0.5*0.57
    4   R     W     W  --->  W             -

    5   W     R     R  --->  R             +                0.5*0.5*0.57
    6   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 

    0.5*0.5*0.57 +0.5*0.5*0.57 = 0.535.       <U>ANSWER</U>
</pre>

Solved.



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