Maths and IQ problems

[littlebigone]

Imagine a graph that plots the position of the monk on the path up the mountain with respect to time.

Plot any random nondecreasing line for the path up the mountain. Then plot any random nonincreasing line going down.

We see that the two lines will cross.

so the monk will be at the same spot at the same time while descending at that spot will be represented by the position that the two lines cross.

[youngyew]

Originally Posted by SpRInG

chiunlin, dun understand ur question lar

1. Let's say you climbed this hill yesterday at 12 pm, using a particular path.

2. You stayed at the mountain peak for the night.

3. On the next day, you started to come down the hill at 12 pm, using exactly the same path.

4. Prove that there exist one point of the path, where you were also at the same place the day before, at the same time.
(e.g. you were half way through the ascend at 3 pm yesterday, and at 3 pm today you were also half way through the descend. So you are at the middle of the mountain, at the same time both day. But it can be anywhere else, and anytime else; you just have to prove that there exist such a point.)

Originally Posted by bp_ffei

I think you should add assumptions, like he is travelling with constant speed etc.

No, he doesn't have to travel with constant speed. The time of the journey also need not be the same. He only has to start both journey at the same time.

[SpRInG]

oh oh
thanks.. i know what the question means now meaning prove that there is a same point where the monk cross at the same time. okay, i think i've got the idea

[bp_ffei]

Originally Posted by youngyew

No, he doesn't have to travel with constant speed. The time of the journey also need not be the same. He only has to start both journey at the same time.

Yup, you're right. How silly of me!! Haha... Good explaination littlebigone!

[windy_city]

Here an interesting problem:
Real life question involving real people:
*For those who know the answer, just post your answer but do not explain why, so that others can try it out too.
The correct answer will be reveal later this week.

Suppose it is assumed that about 5% of the general population use drugs. You employ a test that is 95% accurate, which we?ll say means that if the individual is a user, the test will be positive 95% of the time, and if the individual is a nonuser, the test will be negative 95% of the time. A person is selected randomly and given the test. It is positive. What does such a result suggest? Would you conclude that the individual is highly likely to be a drug user?

4 PHD got the answer wrong when they wrote their answer to the newspaper, a similar question like this was directed to a group of experienced doctor and only 2 doctors (8%) got the correct answers.

This question was given by Prof Scot McKeon in his first class, he is the Kellogg's reigning L. G. Lavengood Outstanding Professor of the Year. All MBA students are required to take this decision analysis class. Well, I think this class is one of the most interesting classes I have taken so far.

[chiunlin]

ok, after thinking for a minute, I think the person is highly likely to be a drug user. about My head is still spinning from doing digital logic design, I'll try and give it a more thorough thought later.

Added five minutes later: The precision of the experiment is independent of the assumption about the population, so since the person is tested positive, the person is 95% likely to be a drug user.

Added yet another two minutes later: Imagine there's a disease that infects 0.00001% of the population and an experiment to test it has a precision of 95%. You randomly pick a person from the street and he's tested positive. Now, is the person highly likely to be carrying the disease? A doctor never takes into account the possiblity of a disease infecting a person, all that matters is the result and the precision of the experiment carried out to test it.

[littlebigone]

I think this has to do with conditional probabilty.

Define events:

Ne = negative test
Po = positive test
D = drug user
ND = non-drug user


P(Po|D) = 0.95
P(Ne|ND) = 0.95
P(D) = 0.05
P(ND) = 0.95

All the above is given in the question

We are interested in P(D|Po)

P(D|Po) = P(Po and D)/P(Po)

P(Po|D) = P(D and P)/P(D)
=> P(D and P) = P(Po|D) * P(D) = 0.95 * 0.05 = 0.0475

Po = (Po and D) union (Po and ND)
=> P(Po) = P(Po and D) + P(Po and ND)
= P(Po|D)P(D) + P(Po|ND)P(ND)
= 0.95*0.05 + 0.05*0.95
= 0.095

Therefore P(D|P) = 0.0475/0.095 = 0.5!!!!

The probability of a person being a drug addict given that his test is positive is 0.5 which is not highly likely.

[windy_city]

So far nobody got it right yet.
But good progress, let more people try it and I will reveal the answer soon.

[chiunlin]

Since it's not very likely nor unlikely, then the answer must be there's an equal chance that the person is a drug addict. Anyway, don't reveal the answer so soon.

[youngyew]

Originally Posted by windy_city

Suppose it is assumed that about 5% of the general population use drugs. You employ a test that is 95% accurate, which we?ll say means that if the individual is a user, the test will be positive 95% of the time, and if the individual is a nonuser, the test will be negative 95% of the time. A person is selected randomly and given the test. It is positive. What does such a result suggest? Would you conclude that the individual is highly likely to be a drug user?

Hmm.. for sure my intuitive thought told me that there is a 95% of chance that the individual is a drug user.

But later on, I changed my mind. Here's my attempted reasoning: (sorry for providing reasoning, but those who want to try themselves can just skip all the reasonings)

SPOILER WARNING

1. When you choose an individual randomly, there's 5 % chance of getting a drug user and 95% chance of getting a nonuser.

2. Users and nonusers are mutually exclusive.

*3.1.1 For users, since 95% of the time we get right result, that means there are 4.75% of population that will show positive results and IS drug user.

*3.1.2 For users, since 5% of the time we get wrong result, that means there are 0.25% of the population that will show negative results but IS drug user.

*3.2.1 For non-users, since 95% of the time we get right result, there are 90.25% of the population that will show negative result and IS NOT drug user.

*3.2.2 For non-users, since 5% of the time we get wrong result, there are 4.75% of population that will show positive results but IS NOT drug user.


Conclusion: Now refer to 3.1.1 and 3.2.2. When you get positive result, there is actually an equal chance of getting a user or nonuser.

If my answer is wrong, can anyone kindly point out which part of my reasoning is wrong.

p/s: I am a medical student.