Are you able to calculate the odds on this?

[mlcpa51]

I cannot find a way to calculate this....... I have 2 candidates being interviewed by 5 different interviewers. Each candidate is asked 13 questions worth varying points. 1 question is worth 20 points, 1 question is worth 10 points, 3 questions are worth 7 points each, 6 questions are worth 5 points each and 2 questions are worth 2 points each. The total possible score from each interviewer is 85 points (20 + 10 + 7 + 7 + 7 + 5 + 5 + 5 + 5 + 5 + 5 + 2 + 2). I then throw out the high and low score from the 5 interviewers for each candidate, leaving 3 scores of a possible 85 each which would give each candidate a possible 255 (85 x3) total score. What are the odds that the 2 candidates would end up with a tied score?

 

[Steven G]

I guess that you failed to read the posting guidelines. If you had you would know that we offer help on this forum and not solutions. If you had followed the guidelines by showing us your attempt at this problem you would have received help by now. So please show us your work, tell us where you are stuck so that we can get you back on track to arriving at the correct answer.

Here is a hint: Can you list all the possible scores a candidate could get?

 

[mlcpa51]

I am so sorry, I did read the posting guidelines, but I am stuck at even how to get started because there are so many different answers or combination of answers that could add up to the same score. If there were just 2 or 3 questions would the chances of the same score not just be a multiple of the possibility of each score? It is when you throw in so many different possible answers to each individual question that I do not know how to approach it. I have searched for an example for so many outcomes and cannot find one. My intent was not to not follow the rules, my intent was to just ask for some help trying to understand where to start.

 

[mlcpa51]

My thinking is because each question can have multiple scores, the number of possible scores would be 20 x 10 x 5 x 5 x 5 x 5 x 5 x 5 x 7 x 7 x 2 x 2 or 4,287,500,000 possible scoring combinations that would add up to 85. That number seems too high. Am I missing something? If it were correct, how do I approach the odds of two people reaching the same score?

 

[HallsofIvy]

"Scoring combinations" is not relevant. It is the total score that is relevant to being tied. The maximum score from each interviewer is 85 points and there are 5 interviewers so the maximum total score for each candidate is 5(85)= 425. Each candidate can have a total score for all 5 interviewers from 0 to 425, 426 different scores. If every score is equally likely (unlikely but we have to assume that to make this problem doable) then there are (426)(426) possible pairs of scores, 426 of them, (0, 0) to (425, 425), are "tied" so the probability of a tie is 426/((426)(426))= 1/426.

 

[JeffM]

I have a question about how this process is supposed to work. What is the definition of a tie? Does that mean all five interviewers agree, three interviewers agree, or two agree? If two agree on the highest score, are both those scores thrown out? If three agree on the lowest score, are all three thrown out. Because I don't understand how the process is supposed to work, I can't even begin to analyze this.

Here is how I would analyze the much simpler problem of a three-way tie among three interviewers without the complications of discarding two out of five results.

First, I would not make HOI's arbitrary and highly implausible assumption that the scores are completely uncorrelated. Instead, I would make the much more plausible qualitative assumption that the scores are highly correlated. THEN I would make my own arbitrary quantitative assumption about what highly correlated means numerically in this specific case. I shall assume that it means that the difference between high and low scores will not exceed 10, but within this range scores are uncorrelated. (On a scale of 85, this is a relatively high spread.) Why do I assume that the scores will cluster. Because three interviewers are interviewing the SAME candidate; presumably, the qualities of the candidate affect the scores in a related way. It is easy to change this quantitative assumption so if you don't like it you can change it.

Say the low score is x and x < 76. Then there is a 1/11 chance that another interviewer will give the same score, but the chance that both the other interviewers will give the same score is (1/11) * (1/11) = 1/121. In other words, the probability of a three way tie is a bit under 1%.

What happens if we decide after looking at the results of many interviews that my guess at clustering was too broad and a better estimate is a difference of 5 between high and low scores. Then the probability of a three-way tie is
(1/6)(1/6) = 1/36, which is under 3%. So our analysis is not very sensitive to the quantitative assumption about spread.

Notice that my estimated probability of a tie is much higher than HOI's estimate. This is because he is assuming no tendency to clustering, and I am assuming a strong tendency to clustering. Different assumptions lead to different conclusions. But also notice that both assumptions estimate a probability of an exact tie as being relatively low.

 

[mlcpa51]

Definition of a tie - each of the 5 interviewers gives each of the candidates (2) a score based on point value with a maximum score from each interviewer of 85. Taking those five scores of the 5 interviewers (which scored each individual candidate) and throwing out the high score and the low score of the five; you then total the 3 remaining scores for each candidate (remember an individual score can be 85) making a possible total score for each candidate 255 (85 x3). The 2 total scores (which can be up to 255) for the individual candidates is what is tied. For instance, each candidate scored 200 out of a possible 255 points. I am trying to figure out the chances if that happening given so many different possible outcomes.

Also remember the 85 possible points given by each interviewer to each of the two individual candidates comes from 13 questions of varying point values.

 

[JeffM]

Definition of a tie - each of the 5 interviewers gives each of the candidates (2) a score based on point value with a maximum score from each interviewer of 85. Taking those five scores of the 5 interviewers (which scored each individual candidate) and throwing out the high score and the low score of the five; you then total the 3 remaining scores for each candidate (remember an individual score can be 85) making a possible total score for each candidate 255 (85 x3). The 2 total scores (which can be up to 255) for the individual candidates is what is tied. For instance, each candidate scored 200 out of a possible 255 points. I am trying to figure out the chances if that happening given so many different possible outcomes.

Also remember the 85 possible points given by each interviewer to each of the two individual candidates comes from 13 questions of varying point values.

So if all five give the same score, do you throw them all out? Or if two score 80 and three score 2, do you throw them all out? Your process is not well defined. You have simply ignored my previous question.

So is it the total score which defines a tie, or the individual scores for each question? Because if it is the total score, there are only 86^5 possibilities. If you assume as Halls did that there is no clustering, the probability that five scores are the same is 86(1/86^5 = 1/86^4, which is effectively zero. The probability that four scores are the same is 86(1/86^4)*85(1/86)*5 = 425/86^4, which is also effectively 0. The probability that three scores are the same is 86(1/86^3)*{85(1/86)}^2*10 = 10*85^2/86^4, which is about 0.1%. So the probability that at least three tie is far less than 2 chanes out of a thousand.

That might get you a good mark in a school room exercise. If, however, this problem has some practical application, you need to understand that the reasoning in the preceding paragraph asumes that each interviewer's score is completely random and has no relation whatsoever to the candidate being interviewed. Based on my experience of Human Resource departments, the assumption that those people make decisions by rolling dice is at least semi-plausible, but if you have competent interviewers, the scores will not be random. In that case, the probability of ties is much higher than what was estimated in the previous paragraph. To get any numerical estimate consistent with competent interviewers, you need to make some sensible assumption about the degree of clustering expected.