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Tags math puzzles , probability puzzles

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Old 17th April 2015, 06:52 AM   #1
Myron Proudfoot
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What are the Odds?

Hi all, I have a math question. I am grading exams for my upper division history class and a student came to me concerned because they thought the two students in front of them were cheating. The first student would tap on something on their test. Student 2 would nod and write something down. The reporting student was concerned they were cheating. Sounds like it to me as well.

The test had a section of short IDs and an essay section. Itís the short ID section that concerns me. There were 27 terms to chose from and students had to pick 10 and write a few sentences on each.

I compared the two studentsí answers and they answered seven of the same 10 IDs AND they were in the same sequence.

If I were to lay out their answers, they would look like this.

Student 1: A, B, C, D, E, F, G, H, I, J

Student 2: M, N, A, C, D, E, F, G, H, I,

Their answers are not exactly alike, but similar. The first studentís answers could easily be seen as paraphrases of the other studentís. FWIW, student 2 majors in history and gets mostly As. Student 1 is outside her major and earned a D+ on her last exam.

My question, all else being equal, what are the odds that two students would pick 7 out of 10 IDs to answer, in the same order, out of a total pool of 27? I know there are other factors that would influence a studentís choice, and they both picked some of the more popular terms. But, what are the odds that their tests are so similar?
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Old 17th April 2015, 07:08 AM   #2
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I think you are correct in discounting that the answers are similar - so long as they are the correct answers, since all correct answers would seem similar.

The probability isn't neutral either. You need to compare with what other test takers have done. There may be some outside reason why those are selected in a particular order. For example, if I were taking the test, I might try the easiest first - the ones I have the most confidence in.

What's needed here, and what will sell you on cheating, is a comparison with others in the class as a control group. Only then will the raw probability be convincing.

Unfortunately, you may discover that others in the class are cheating as well - something that would taint the control group. But the important thing is the existence of missing variables in your analysis.

Off the cuff, I would say the evidence favors cheating, partly because of the addition of witness testimony and partly because of the status of purported offenders (A vs D student).

A re-test under controlled conditions should clarify matters. Just eliminate the suspect answers already given from the pool of 27 and test each student separately. Framed as a chance to avoid a charge of cheating, I think the students would go for it.
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Old 17th April 2015, 07:15 AM   #3
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I agree with the above post. Having everyone retake the test, possibly with random seating added, is probably the least confrontational way of dealing with the situation.

In order to not punish the person who came to you about this, maybe offer them the higher of the two test scores as well. Otherwise they may feel that next time the see cheating happen, they won't tell anyone because they don't want to have to retake a test again.
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Old 17th April 2015, 07:34 AM   #4
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would it help if I could give a total for each time a particular term was ID'd?
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Old 17th April 2015, 08:16 AM   #5
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How in the world would tapping on an item on a test with several-sentence written answers convey the correct (or any) answer? I assume that the student wasn't describing a minute or two of continuous tapping of tap code or Morse code for each question. Were the test answer sheets visible to students at adjacent desks (and if so, why bother with the tapping/nodding)?

Also, was it seven of the items in common, or eight as your letter lists suggest?

The straightforward probability calculation you're asking for (chance of getting 7 matches with 10 targets when choosing 10 items from a pool of 27) is rather complex. The chance of getting 7 matches to 10 targets when choosing 7 items from a pool of 27 is easily calculated as (10/27)*(9/26)*...*(4/17) = .000135, but having 10 choices will considerably increase that chance, and if some of the choices are more attractive, the chance could rise above a level sufficient for suspicion. On the other hand, the match in sequence is also highly improbable. But on the gripping hand, the match in sequence is not improbable if the A, B, C, etc. also represents the order the items were listed in the test.

Also, may I ask if this is a college history class (as "upper level" suggests) or grade/high school? (To me, whether or not the students are there by choice makes a big difference in how I perceive the morality of such "cheating.")
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Old 17th April 2015, 10:16 AM   #6
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From your description it appears that the direction of (potential) communication was from the D student to the A student. I'm not sure how that would facilitate cheating.
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Old 17th April 2015, 10:28 AM   #7
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I'm not sure you could prove anything from the evidence at hand and I don't think it's right to penalize a student even on a strong suspicion from the other student and the test results.

Often you can catch cheaters more by wrong answers than by correct answers.

If you really want to see if they are cheating, create a multiple choice quiz with maybe 20 - 25 answers. Have two versions with all the same questions but the answers in differing order for some of the questions. Make sure those two students each get a different version of the test. If student two is cheating they will just write down A, B, C or D without really reading the answers.
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Old 17th April 2015, 11:28 AM   #8
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Originally Posted by Spindrift View Post
I'm not sure you could prove anything from the evidence at hand and I don't think it's right to penalize a student even on a strong suspicion from the other student and the test results.

Often you can catch cheaters more by wrong answers than by correct answers.

If you really want to see if they are cheating, create a multiple choice quiz with maybe 20 - 25 answers. Have two versions with all the same questions but the answers in differing order for some of the questions. Make sure those two students each get a different version of the test. If student two is cheating they will just write down A, B, C or D without really reading the answers.
Back in the old days when we teachers could make our own tests, I used to do that last.................for the entire class.
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Old 18th April 2015, 05:47 AM   #9
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Originally Posted by Myron Proudfoot View Post
Hi all, I have a math question. I am grading exams for my upper division history class and a student came to me concerned because they thought the two students in front of them were cheating. The first student would tap on something on their test. Student 2 would nod and write something down. The reporting student was concerned they were cheating. Sounds like it to me as well.

The test had a section of short IDs and an essay section. It’s the short ID section that concerns me. There were 27 terms to chose from and students had to pick 10 and write a few sentences on each.

I compared the two students’ answers and they answered seven of the same 10 IDs AND they were in the same sequence.

If I were to lay out their answers, they would look like this.

Student 1: A, B, C, D, E, F, G, H, I, J

Student 2: M, N, A, C, D, E, F, G, H, I,

Their answers are not exactly alike, but similar. The first student’s answers could easily be seen as paraphrases of the other student’s. FWIW, student 2 majors in history and gets mostly As. Student 1 is outside her major and earned a D+ on her last exam.

My question, all else being equal, what are the odds that two students would pick 7 out of 10 IDs to answer, in the same order, out of a total pool of 27? I know there are other factors that would influence a student’s choice, and they both picked some of the more popular terms. But, what are the odds that their tests are so similar?
If it's truly a random pick of questions, then the odds are pretty low. I note that the 7 identical questions are moreover consecutive.

Number of ways to pick 10 ordered questions: 27 * 26 * 25 ... 18

Number of ways for student 2 to pick the first 7 questions identical to the first 7 questions of student 1, and then questions #8-#10 differently: the only choice now is in choosing the different questions, which gives 17 * 16 * 15 possibilities. But student 2 could also pick his questions #1-#7 as identical to student 1's questions #2-#8, or #3-#9, or #4-#10. That gives 4 different ways to organize them. The identical questions could also be #2-#8 on student 2's answer sheet, with again 4 different sequences on student 1's answer sheet, etc. In total, 16 different ways of picking the 2 sequences.

So, in total we have: 17 * 16 * 15 * 4 * 4.

If the questions need not be consecutive, you get 17 * 16 * 15 * C(10,3) * C(10,3) where C is the binomial coefficient: it's the number of ways to pick the sequence number of the non-identical questions.

ETA:
So the final probability (in the non-consecutive case) would be:
( 17 * 16 * 15 * C(10,3) * C(10,3) ) / ( 27 * 26 * 25 * ... * 18 )

But there comes a lot more into it, most importantly: the questions aren't evenly popular, some will have been picked by about everyone.

And as noted above, the mechanism of cheating is not quite clear. Student 1 must have been the cheater here, not student 2. The only way the tapping makes sense if it were a kind of command by student 1 for student 2 to select and write down the answer to a specific question, so student 1 then could have cheated and copied the answer. That seems a bit outrageous. Moreover, "similar" doesn't quite cut it then, IMHO.
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Old 18th April 2015, 10:25 PM   #10
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Originally Posted by Myriad View Post
How in the world would tapping on an item on a test with several-sentence written answers convey the correct (or any) answer? I assume that the student wasn't describing a minute or two of continuous tapping of tap code or Morse code for each question. Were the test answer sheets visible to students at adjacent desks (and if so, why bother with the tapping/nodding)?

Also, was it seven of the items in common, or eight as your letter lists suggest?

The straightforward probability calculation you're asking for (chance of getting 7 matches with 10 targets when choosing 10 items from a pool of 27) is rather complex. The chance of getting 7 matches to 10 targets when choosing 7 items from a pool of 27 is easily calculated as (10/27)*(9/26)*...*(4/17) = .000135, but having 10 choices will considerably increase that chance, and if some of the choices are more attractive, the chance could rise above a level sufficient for suspicion. On the other hand, the match in sequence is also highly improbable. But on the gripping hand, the match in sequence is not improbable if the A, B, C, etc. also represents the order the items were listed in the test.

Also, may I ask if this is a college history class (as "upper level" suggests) or grade/high school? (To me, whether or not the students are there by choice makes a big difference in how I perceive the morality of such "cheating.")
I initially agree with this. Reading a short sentence and paraphrasing it might work for one answer but it'd be damned hard for 7. It would be pretty apparent to anyone monitoring the test that 1 was making an effort to read 2's work.

But... then it occurred to me.... are there the equivalent of simple 1.5x to 3x contact lenses? Like reading glasses? For some of the higher multiples, they would screw up the reading of a book for me, but I could see things much clearer that were a meter away. And there's no mention if 1 wore glasses. You wouldn't need the contacts if you just had a powerful set of reading glasses. And writing with them on is easy; just lower your gaze so you're looking under the lenses.
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Old 20th April 2015, 05:08 AM   #11
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Originally Posted by Myriad View Post
How in the world would tapping on an item on a test with several-sentence written answers convey the correct (or any) answer? I assume that the student wasn't describing a minute or two of continuous tapping of tap code or Morse code for each question. Were the test answer sheets visible to students at adjacent desks (and if so, why bother with the tapping/nodding)?


Also, may I ask if this is a college history class (as "upper level" suggests) or grade/high school? (To me, whether or not the students are there by choice makes a big difference in how I perceive the morality of such "cheating.")
Thanks.
1. The D student was asking the A student to answer that particular question. The D student could then paraphrase the A student's answer. As for why they were visible, we have 2 person tables. I don't have any choice but to sit them next to one another. In my survey classes they get alternative tests so people sitting next to one another can't copy. I may have to do that here as well.

2. College.
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Old 20th April 2015, 05:10 AM   #12
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Originally Posted by Spindrift View Post
I'm not sure you could prove anything from the evidence at hand and I don't think it's right to penalize a student even on a strong suspicion from the other student and the test results.

Often you can catch cheaters more by wrong answers than by correct answers.

If you really want to see if they are cheating, create a multiple choice quiz with maybe 20 - 25 answers. Have two versions with all the same questions but the answers in differing order for some of the questions. Make sure those two students each get a different version of the test. If student two is cheating they will just write down A, B, C or D without really reading the answers.
I use multiple choice in my survey classes for the very reason you cite. Sometimes it's the exact same question but I switch the correct answer with a wrong one. However, I use essay tests in my upper division classes. I want to see them discuss what they learned (or didn't) not just tell me they remembered what the Zimmerman telegram was... We are also under a lot of pressure here not to use ANY multiple choice.
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Old 20th April 2015, 05:19 AM   #13
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thanks for all the feedback. I am still debating what, if anything, to do, and will be meeting with my department chair later to discuss options. Given the lack of definite proof I'll probably read the riot act to both students, put a little fear of God into them, and assign seats for the final. I'll also be copying what some of my colleagues do and sit behind the class, rather than in front of it, so they can not see if I am watching them. I usually walk around the classroom at random times during the test, but that always distracted me when I was a student, so I did not do it for the whole class... Ugh.
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Old 24th April 2015, 06:44 AM   #14
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FYI, I brought the students into my office and they admitted they cheated the way I suggested. They will get Fs on the exam but that is all. Thanks for all the feedback..
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