superprismatic Posted April 12, 2011 Report Share Posted April 12, 2011 The University of Southern North Carolina at Hoople wants to estimate the number of its students who have experimented with illegal drugs at some point in their lives. But, such drug experimentation is illegal and subject to severe punishment. So a polling procedure was designed such that an admission of guilt during the poll would not be sufficiently convincing for a person to be found guilty of this offence. Here are the details of the procedure which USNCH has instituted and which each of USNCH's 10,000 students is required to take: 1. The student is placed in a secluded room where there are 100 marbles in 4 lines of 25 each arrayed on a board which has been dimpled to hold the marbles in this configuration. Three lines contain green marbles and one line contains red marbles. The student is supplied with an empty black bag into which he places all the marbles. 2. The student has been told to randomize the bag (by shaking it), reach in and randomly pick one marble, and surreptitiously peek at it to determine its color. 3. The student is told to tell the truth if the marble he picked is green, but to lie if the picked marble is red. 4. The student is asked the question, "Have you ever experimented with illegal drugs?" He is to answer according to step 3. 5. The answer is tallied with those of all the other students. It is known (I don't know how) that 75% of the students will follow the instructions correctly. It is also known that 10% of the student body (but their identities are unknown) are members of the Verititian religion who always tell the truth when asked a question, regardless of any other circumstances. A further 10% are anarchists who, out of sheer contempt for authority, will always do the opposite of what they are told -- in this case they would tell the truth if they saw a red marble and lie if they see a green one. The remaining 5% belong to the Nastyranian cult and will always lie regardless of the marble color picked. Which students are in which of these four groups is unknown. 4,625 of the students answered the question in the affirmative. What is the most likely number of students who have actually experimented with illegal drugs? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted April 12, 2011 Report Share Posted April 12, 2011 just for clarification, that is 4.625%? or is it 4 and 5/8ths of a person? Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted April 12, 2011 Author Report Share Posted April 12, 2011 just for clarification, that is 4.625%? or is it 4 and 5/8ths of a person? Neither, it is 4,625 students out of 10,000 students. That would be 46.25% of the student body. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted April 12, 2011 Report Share Posted April 12, 2011 (edited) Number of people that follow directions = 7500 (5625 tell the truth and 1875 lie) Number of people that always tell the truth = 1000 Number of people that do the opposite of directions = 1000 (250 tell the truth and 750 lie) Nuber of people that always lie = 500 So 6875 people tell the truth and 3125 lie. Since 4625 said yes it can be assumed that 68.75% actually experimented of that group. Of the people that said no (5375) it can be assumed that 31.25% (the liars) actually experimented So the number of people is 4625*0.6875 + 5375*0.3125 = 4859 Edited April 12, 2011 by ArtDigby Quote Link to comment Share on other sites More sharing options...
0 moonthirsty Posted April 12, 2011 Report Share Posted April 12, 2011 I'm not going to bother with the math diversion. The answer is 100%. Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted April 12, 2011 Author Report Share Posted April 12, 2011 Number of people that follow directions = 7500 (5625 tell the truth and 1875 lie) Number of people that always tell the truth = 1000 Number of people that do the opposite of directions = 1000 (250 tell the truth and 750 lie) Nuber of people that always lie = 500 So 6875 people tell the truth and 3125 lie. Since 4625 said yes it can be assumed that 68.75% actually experimented of that group. Of the people that said no (5375) it can be assumed that 31.25% (the liars) actually experimented So the number of people is 4625*0.6875 + 5375*0.3125 = 4859 I think you assumed too much. If we assume your answer (that 4,859 of the 10,000 actually experimented), then our best guess for the probability of a student having experimented is .4859. If we use this to compute the expected number of students who answer yes in each of the four cases, they add up to 4,947.075 which is much bigger than the 4,625 stated in the problem. Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted April 12, 2011 Author Report Share Posted April 12, 2011 I'm not going to bother with the math diversion. The answer is 100%. [spoiler=I see what you're driving at. ]But, the math would not work out to having 4,625 students saying yes. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted April 12, 2011 Report Share Posted April 12, 2011 Number of people that follow directions = 7500 (5625 tell the truth and 1875 lie)Number of people that always tell the truth = 1000Number of people that do the opposite of directions = 1000 (250 tell the truth and 750 lie)Nuber of people that always lie = 500So 6875 people tell the truth and 3125 lie.Since 4625 said yes it can be assumed that 68.75% actually experimented of that group.Of the people that said no (5375) it can be assumed that 31.25% (the liars) actually experimentedSo the number of people is 4625*0.6875 + 5375*0.3125 = 4859 Same number truth and lie tellers (6875 and 3125) Then I didn't follow the assumptions, sorry. Instead, I solved for p, the proportion who have experimented: p*(6875)+(1-p)*3125=4625 p=0.4 of the true population, if everyone told the truth, would say yes. 0.4*10,000 = 4,000 people Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted April 13, 2011 Author Report Share Posted April 13, 2011 Same number truth and lie tellers (6875 and 3125) Then I didn't follow the assumptions, sorry. Instead, I solved for p, the proportion who have experimented: p*(6875)+(1-p)*3125=4625 p=0.4 of the true population, if everyone told the truth, would say yes. 0.4*10,000 = 4,000 people You've got it! 4,000 is correct and, I might add, you gave a very simple explanation. Quote Link to comment Share on other sites More sharing options...
0 Peekay Posted April 13, 2011 Report Share Posted April 13, 2011 (edited) You've got it! 4,000 is correct and, I might add, you gave a very simple explanation. From the info given, we can get the following Truth Table: Truth Lie 75% 7500 5625 1875 10% 1000 0000 1000 10% 1000 0250 0750 05% 0500 0000 0500 # 10000 6875 3125 So, 68.75% of the students will say the truth while the rest will lie. Given that 4,625 students said YES, 68.75% of that number, that is 3,180 students actually did. Out of the rest of the students (who said NO), 31.25% would be lying, that is 1,680 would actually have. In total 4,860 students are likely to have experimented with illegal drugs. Edited April 13, 2011 by Peekay Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted April 13, 2011 Author Report Share Posted April 13, 2011 From the info given, we can get the following Truth Table: Truth Lie 75% 7500 5625 1875 10% 1000 0000 1000 10% 1000 0250 0750 05% 0500 0000 0500 # 10000 6875 3125 So, 68.75% of the students will say the truth while the rest will lie. Given that 4,625 students said YES, 68.75% of that number, that is 3,180 students actually did. Out of the rest of the students (who said NO), 31.25% would be lying, that is 1,680 would actually have. In total 4,860 students are likely to have experimented with illegal drugs. ArtDigby did the same thing. In my spoiler in post #6, I said that if you use the same number you came up with, and use it to compute how many affirmative answers you get, the number is way higher than 4625. Try it yourself. Quote Link to comment Share on other sites More sharing options...
Question
superprismatic
The University of Southern North Carolina at Hoople
wants to estimate the number of its students who have
experimented with illegal drugs at some point in
their lives. But, such drug experimentation is illegal
and subject to severe punishment. So a polling
procedure was designed such that an admission of
guilt during the poll would not be sufficiently
convincing for a person to be found guilty of this
offence. Here are the details of the procedure
which USNCH has instituted and which each of USNCH's
10,000 students is required to take:
1. The student is placed in a secluded room
where there are 100 marbles in 4 lines of
25 each arrayed on a board which has been
dimpled to hold the marbles in this
configuration. Three lines contain green
marbles and one line contains red marbles.
The student is supplied with an empty black
bag into which he places all the marbles.
2. The student has been told to randomize the
bag (by shaking it), reach in and randomly
pick one marble, and surreptitiously peek
at it to determine its color.
3. The student is told to tell the truth if the
marble he picked is green, but to lie if the
picked marble is red.
4. The student is asked the question, "Have
you ever experimented with illegal drugs?"
He is to answer according to step 3.
5. The answer is tallied with those of all the
other students.
It is known (I don't know how) that 75% of the students
will follow the instructions correctly. It is also
known that 10% of the student body (but their identities
are unknown) are members of the Verititian religion
who always tell the truth when asked a question,
regardless of any other circumstances. A further
10% are anarchists who, out of sheer contempt for
authority, will always do the opposite of what they
are told -- in this case they would tell the truth if
they saw a red marble and lie if they see a green one.
The remaining 5% belong to the Nastyranian cult and
will always lie regardless of the marble color picked.
Which students are in which of these four groups is
unknown.
4,625 of the students answered the question in the
affirmative. What is the most likely number of students
who have actually experimented with illegal drugs?
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