OK - I think we're getting close (thanks, Stan & Tim). There are four "innocent" outcomes - 1111, 2222, 3333, and 4444. But there are 256 total possibilities (4 four each of 4 places, 4*4*4*4 = 256), so the p "innocence" by chance = 4/256 = .0156.
I agree a Bayesian approach would be useful, since we really want the p of innocence or guilt, not the p of the data being such-and-such. Using the "odds" ratio for of Bayes theorem (H = alternate hypothesis, ~H = null, D = data): posterior odds = prior odds * relative likelihood p (H / D) p (H) p (D / H) -------- = ----- * --------- p (~H / D) p (~H) p (D / ~H) H = they are innocent and ~H = they lie, D = all 4 students getting the same tire. p(H) and p (~H)? Let's say "even odds" - 1/1 p (D / H) = getting all 4 tires the same if innocent. Close to 1, but let's say = .90 since memory is imperfect. p (D / ~H) = getting all 4 tires by chance = 4/256 = .015 So, posterior odds = (1/1) * (.9/.015) = or 60 to 1 in favor of innocence, or p innocence = 60/61 = .98. If the professor had other evidence that increased her intuition they were guilty (past history, for instance), and thought it 2 to 1 they lied, (1/2)* (.9/.015) = 30 to 1 odds of innocence, or p innocence = 30/31 = .968. Doesn't change much unless we really tweak the prior odds. If we make D = getting different tires, p innocence is (1/1) * (.1/.98) or 10 times more likely to be guilty if they mess up the tire question (p guilt =.91). If prior odds in favor of guilt are 2 to 1, then (1/2) * (.1 /.98) = 20 to 1 in favor or guilt (p guilt = .95). (I did these ## quickly - are they correct??). Tim: this is totally ignoring your point which is the p of different tires being flat. Yikes!! ============================================ John W. Kulig Professor of Psychology Plymouth State College Plymouth NH 03264 ============================================ "Nothing will benefit human health and increase chances for survival of life on Earth as much as the evolution to a vegetarian diet" - Albert Einstein > -----Original Message----- > From: Shearon, Tim [mailto:[EMAIL PROTECTED] > Sent: Wednesday, December 01, 2004 5:08 PM > To: Teaching in the Psychological Sciences > Subject: RE: Final Exam > > > Stan - Depends. What is the 4 in 4 x 4 x 4 to represent? Seems to me that > there is a .25 chance (1/4) for each one to pick each tire. That yields a > .25 x .25 x .25 = 0.015625 or 1.5625% chance of innocence. But is that > really correct. I think the actual number of events is really more > complicated that it might seem. I'm assuming that anyone in college taking > a test has some experience with flat tires on their car. So one way to > approach it would require a Bayesian knowldege of the history of those > individuals (to know what bias would seed their choices- for me it is > passenger side front hands down!). I know, for example, that the overall > rate of flats is higher on some tires than others (drivers side vs > passenger, front vs rear, etc.). That's why they tell us to rotate the > tires is to even out the wear etc. Course, who actually does that on our > busy schedules. I think you could work out any number of possible senarios > but all of them would mean that the three aren't very likely to guess > correctly. Their only "chance" of passing were anyone dumb enough to do > this is to have worked their response out in advance. It is still a > humorous idea even though I bet this is urban myth. Kind of like the > "graduate student proctoring an exam and sees a student cheating. She > waits till the student is finished and begins to turn in their answer > sheet (blue book, etc- depends on the version). She tells the student to > keep their sheet as she has seen them cheating. The student looks bemused > for a moment then says, 'Do you know who I am?' After a heated exchange > the grad student says, 'No I don't and it doesn't matter.' At which point > the student puts their paper in the stack, throws the papers into the air > and exits the room." These are all cute and funny but probably never > happened anyway. :) > Tim Shearon, Albertson College of Idaho, Caldwell, ID > > -----Original Message----- > From: Stanley Cohen [mailto:[EMAIL PROTECTED] > Sent: Wed 12/1/2004 1:35 PM > To: Teaching in the Psychological Sciences > Cc: > Subject: RE: Final Exam > John, > I calculate an even higher "chance rate." There 4x4x4=64 events in the > outcome space. In four of the events (assume tires labeled 1,2,3,4)-- > (1,1,1), (2,2,2),(3,3,3),(4,4,4)--the jury finds them innocent. Thus > 4/64=.0625. > > Is my logic gone astray somewhere? > > Best, > Stan > > > >>>>>>>>>>><<<<<<<<<<<<<< > "If you're going through hell, keep going." > - Winston Churchill > > Stanley H. Cohen, PhD > Professor & Interim Associate Chair > Faculty Senate Chair, 2002-2003 > Department of Psychology > West Virginia University > 1220 Life Sciences Building > Morgantown, WV 26506-6040 > E-Mail: [EMAIL PROTECTED] > http://www.as.wvu.edu/psyc/Faculty/SCohen/index.htm > Tel. No.: 304-293-2001x31641 > Fax: 304-293-6606 > >>>>>>>>>>>>><<<<<<<<<<<< > > >>> [EMAIL PROTECTED] 12/1/2004 10:42:36 AM >>> > > Of course, there is a non-trivial probability of choosing the same tire > by chance. Nearly 16 times out of 1000 (from ¼ * ¼ * ¼) lying students > would all agree on the same tire. Scenes we'd like to see: The jury finds > you INNOCENT, p = .016 J > ============================================ > John W. Kulig > Professor of Psychology > Plymouth State College > Plymouth NH 03264 > ============================================ > -----Original Message----- > From: Gary Klatsky [mailto:[EMAIL PROTECTED] > Sent: Wednesday, December 01, 2004 9:37 AM > To: Teaching in the Psychological Sciences > Subject: FW: Final Exam A cute parable for this time of year Gary J. > Klatsky, Ph. D.Director, Human Computer Interaction M.A. Program > Department of Psychology [EMAIL PROTECTED] State > University (SUNY) http://www.oswego.edu/~klatsky7060 State Hwy 104W > Voice: (315) 312-3474Oswego, NY 13126 Fax: > (315) 312-6330 All of us who are concerned for peace and triumph of reason > and justice must be keenly aware how small an influence reason and honest > good will exert upon events in the political field. > Albert Einstein At Duke University, there were four sophomores taking > Chemistry and all of them had an 'A' so far. These four friends were so > confident that the weekend before finals, they decided to visit some > friends and have a big party. They had a great time, but after all the > hearty partying; they slept all day Sunday and didn't make it back to Duke > until early Monday morning. Rather than taking the final then, they > decided that after the final, they would explain to their professor why > they missed it. They said that they visited friends but on the way back, > had a flat tire. As a result, they missed the final. The professor agreed > they could make up the final the next day. The guys were excited and > relieved. They studied that night for the exam. He placed them in > separate rooms and gave them a test booklet. They quickly answered the > first problem, worth five point! s. Cool, they thought! Each one, in > separate rooms, thought this is going to be easy. Then they turned the > page. On the second page was written: For 95 points: Which tire --- > You are currently subscribed to tips as: [EMAIL PROTECTED] > To unsubscribe send a blank email to [EMAIL PROTECTED] > > --- > You are currently subscribed to tips as: [EMAIL PROTECTED] > To unsubscribe send a blank email to [EMAIL PROTECTED] > > > > --- > You are currently subscribed to tips as: [EMAIL PROTECTED] > To unsubscribe send a blank email to [EMAIL PROTECTED] > > --- You are currently subscribed to tips as: archive@jab.org To unsubscribe send a blank email to [EMAIL PROTECTED]
