Question 1200745: 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])
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 0.535
This value is exact without any rounding done to it.
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Explanation:
Let's give the judges code names: {J1,J2,J3}
We'll have them be presented in the order that was mentioned in the instructions.
J1 and J2 make their decisions independently of each other.
The probability they reach a correct decision is p = 0.5 for each of those two judges.
If {J1,J2} reach the same outcome, then J3 rules the same way.
If {J1,J2} reach differing outcomes, then J3 thinks for her/himself to reach their own decision (making a mistake with probability q=0.43).
For this situation J3 makes the correct decision with probability 1-q=1-0.43=0.57
Here is a way to represent a two-way table of outcomes for the first two judges.
| J2 right | J2 wrong | | J1 right | | | | J1 wrong | | |
For example, the upper left corner has both J1,J2 give the correct ruling.
Here is a non-tabular format to represent those four outcomes:
J1 right, J2 right
J1 right, J2 wrong
J1 wrong, J2 right
J1 wrong, J2 wrong
Let's list those four outcomes in a new table.
We'll have them as the 4 rows. The 2 columns will be the outcomes for J3.
| J3 right | J3 wrong | | J1 right,J2 right | A | | | J1 right,J2 wrong | B | C | | J1 wrong,J2 right | D | E | | J1 wrong,J2 wrong | | F |
I then added letters A through F to represent the various possible scenarios.
Scenario A is when all three judges make the correct decision.
The cell next to A is blank because J3 doesn't make the wrong decision when her/his two other colleagues make the correct decision (remember J3 mimics the other two judges if they rule the same way). This also explains why we have a blank cell in the bottom left corner.
Scenario B is when J1 gets it right, J2 gets it wrong, J3 gets it right.
Scenario C is when J1 gets it right, J2 gets it wrong, J3 gets it wrong.
And so on.
The letters A, B, and D correspond to situations where the court gets the correct outcome.
This is because there are at least 2 judges that get the correct ruling.
A = all 3 judges get it right
B = J1 and J3 get it right
D = J2 and J3 get it right
All other situations involve at least 2 judges getting the wrong verdict; hence the wrong final outcome.
To find the probability that the court gets the correct decision, we need to calculate the following:
P(A)
P(B)
P(D)
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P(A) = P(J1 right,J2 right)
P(A) = P(J1 right)*P(J2 right)
P(A) = p*p
P(A) = p^2
P(A) = 0.5^2
P(A) = 0.25
This works because J1,J2 are independent of each other.
J3 does not alter the outcome here, so we ignore the probability associated with this judge.
P(B) = P(J1 right, J2 wrong, J3 right)
P(B) = P(J1 right)*P(J2 wrong)*P(J3 right)
P(B) = p*(1-p)*(1-q)
P(B) = 0.5*(1-0.5)*(1-0.43)
P(B) = 0.1425
P(D) = P(J1 wrong, J2 right, J3 right)
P(D) = P(J1 wrong)*P(J2 right)*P(J3 right)
P(D) = (1-p)*p*(1-q)
P(D) = (1-0.5)*0.5*(1-0.43)
P(D) = 0.1425
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Conclusion:
P(A) = 0.25
P(B) = 0.1425
P(D) = 0.1425
P(A)+P(B)+P(D) = 0.25+0.1425+0.1425 = 0.535 represents the probability the court gets the right decision.
The court gets it right exactly 53.5% of the time.
This is around 50%, so it's about as good as a coin toss in my opinion. Those don't seem like good odds.
Answer by ikleyn(52793)  (Show Source):
You can put this solution on YOUR website! .
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.
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
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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. ANSWER
Solved.
This my solution is written as a description of an ALGORITHM
calculating the desired probability based on given input data.
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