[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?

[Guest]

just for clarification, that is 4.625%? or is it 4 and 5/8ths of a person?

[superprismatic]

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.

[Guest]

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

[moonthirsty]

I'm not going to bother with the math diversion. The answer is 100%.

[superprismatic]

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.

[superprismatic]

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.

[Guest]

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

[superprismatic]

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.

[Peekay]

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

[superprismatic]

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.