Probability quiz

[AmeliesDad]

Since you're all gamblers at heart you'll be experts at odds and probabilities and especially the math/stats students will probably already have answered these before. But I think they are great questions to discuss over and over so here we go with a few stats questions:

1) You might know of this as the Monty Hall problem and seen it referenced in mainstream media.
You're on a game show and the host offers you the choice of 3 doors, one has a car behind it and the other 2 have goats (quiet goats). After you have chosen the host opens one of the other 2 doors revealing a goat and asks if you would like to switch. What are the odds of the car being behind each of the 3 doors? And should you switch? (hint: the one he's opened has a 0% chance )

2) This one was new to me this week and involves the same set up, except:
After you select your door the host trips over his mic cable and stumbles through a door at random. It happens to be a door you didn't choose and contains a goat. Now what are the probabilities of the car being behind each door (hint: the one he's opened still has a 0% chance )

3) It is known that in the general population a certain drug is taken my 0.5% of people. A company devises a test that is 99% sensitive and 99% specific (only 1% of positive tests are false positives and the same is true of negative results). A staff member in your office is tested positive but protests his innocence, what is the probability that he is a user?

4) If you continually flip a coin until the sequence HTT arises and then start over you can average your results to find the average number of flips it takes to appear. Then you do the same for the sequence HTH. Which of theses statements is true.
a) The average will be the same
b) The average number for HTT will be lower
c) The average number for HTH will be lower

5) You meet a stranger in the street who tells you that they have 2 children and at least one of them is a girl. What are the odds that their other child is a boy?

Hopefully I haven't left any ambiguity in some of those questions because that would make them infinitely harder to answer

[s0095063]

I've read a bit about the monty hall problem so I know the answers to Qs 1 and 2( i think!).

In a similar vein, i used to think you should always switch on Deal or no Deal when there are only 2 boxes left, but apparently that works on the same basis as Q2 does and so the odds are still 50/50.

[AmeliesDad]

In a similar vein, i used to think you should always switch on Deal or no Deal when there are only 2 boxes left, but apparently that works on the same basis as Q2 does and so the odds are still 50/50.

Giving away even more of the answers here but... Since the answer to all of those Monty Hall type problems is either the odds are the same or you're better off switching then I would switch anyway just in case The problem with Deal or No Deal and, in actual fact the original Monty Hall game show (rather than the question as set), is that if the switch does not *have* to be offered by the host, which it doesn't/didn't, then you are in yet another game state where they may be trying to trick you... Ignore that though and stick to the questions I asked

3 and 4 are the most interesting to me. The women who was jailed for killing her 2 cot death babies and then freed on appeal is kind of relevant to number 3, and is the reason the "expert witness" is being investigated. Genetics and enzymes that cut DNA on certain sequences, etc, are interested in number 4 Its all very interesting (or difficult and boring depending on your outlook).

EDIT: Ooh, I forgot to mention that number 4 is also a subset of a really good hustle

[s0095063]

Giving away even more of the answers here but... Since the answer to all of those Monty Hall type problems is either the odds are the same or you're better off switching then I would switch anyway just in case The problem with Deal or No Deal and, in actual fact the original Monty Hall game show (rather than the question as set), is that if the switch does not *have* to be offered by the host, which it doesn't/didn't, then you are in yet another game state where they may be trying to trick you... Ignore that though and stick to the questions I asked

3 and 4 are the most interesting to me. The women who was jailed for killing her 2 cot death babies and then freed on appeal is kind of relevant to number 3, and is the reason the "expert witness" is being investigated. Genetics and enzymes that cut DNA on certain sequences, etc, are interested in number 4 Its all very interesting (or difficult and boring depending on your outlook).

EDIT: Ooh, I forgot to mention that number 4 is also a subset of a really good hustle

I've read about Q 3 too, so i wont answer that one - it's interesting though. Derren brown has a very similar one in his book when he is describing how easily we can fall for cognitive illusions.

Question 4) answer a) seems the most sensible to me, but i'm guessing that's wrong and i am easy meat for any would-be hustlers

Question 5) I would say 50% but fully expect to be shot down there too!

[Andy]

1 and 2 = Its 50% and you should switch, which I learnt (or first read about) in the book "The Curious incident of the dog in the night" (or similar) a few years ago and thought was fascinating.

3 = 0.5% but I am basing that on the first statement, and disregarding the other two. The first statement is the only one that really matters.

4 = a I would have thought?

5 = 50% still, but again I expect to be wrong as its too obvious.

[munk]

1) You might know of this as the Monty Hall problem and seen it referenced in mainstream media.
You're on a game show and the host offers you the choice of 3 doors, one has a car behind it and the other 2 have goats (quiet goats). After you have chosen the host opens one of the other 2 doors revealing a goat and asks if you would like to switch. What are the odds of the car being behind each of the 3 doors? And should you switch? (hint: the one he's opened has a 0% chance )


2) This one was new to me this week and involves the same set up, except:
After you select your door the host trips over his mic cable and stumbles through a door at random. It happens to be a door you didn't choose and contains a goat. Now what are the probabilities of the car being behind each door (hint: the one he's opened still has a 0% chance )

I was thinking of posting about this the other week as it goes after it was on Horizon on the BBC.

Not entirely sure of the difference between 1 and 2, but generally (according to that horizon show anyway) the paradoxical / counterintuitive answer is that by switching after the first 'losing' choice/door has been revealed you greatly increase your chances of winning (intuitively you would think that it would make no difference at all if you didn't switch because your chance of winning is still 1 in 2). Don't ask me for the maths behind it, I will google it though now you've given it a name! Cheers.

[s0095063]

I was thinking of posting about this the other week as it goes after it was on Horizon on the BBC.

Not entirely sure of the difference between 1 and 2, but generally (according to that horizon show anyway) the paradoxical / counterintuitive answer is that by switching after the first 'losing' choice/door has been revealed you greatly increase your chances of winning (intuitively you would think that it would make no difference at all if you didn't switch because your chance of winning is still 1 in 2). Don't ask me for the maths behind it, I will google it though now you've given it a name! Cheers.

I think the difference lies in the fact that in Q1 the location of the car is known and so the host knows he can't open one of the doors, whereas in Q2 he effectively opens a random door which could've been the car.

[AmeliesDad]

I think the difference lies in the fact that in Q1 the location of the car is known and so the host knows he can't open one of the doors, whereas in Q2 he effectively opens a random door which could've been the car.

That's the crux of it. So in question 1 you had a 33% chance at the start and he knowingly gave you the information needed to now give you the 66% chance if you switch. In the second its a bit more deal or no deal game space and you've resolved down to a 50/50 chance. Bizarre and caused days of chat between me and my colleague here, as well as on forums!

And question 5 is much the same (if I worded it correctly), in that we did not select a 2 child family where there was at least one girl before setting the question. This random person could have had BB, BG, GB or GG, except we've now discarded the BB situation leaving us 2 boys from 3 as the other. 66%.

Lets save some time on that one - Math Forum - Ask Dr. Math

[AmeliesDad]

3 = 0.5% but I am basing that on the first statement, and disregarding the other two. The first statement is the only one that really matters.

Then the test would have been completely useless rather than just slightly useless. It's a pretty well know fact in the medical world that a 99% sensitive test isn't particularly good if the thing you are testing for is rare. The result of the test actually depends on how rare the thing you are testing for actually is! So not 0.5% but not 99% either. Any other answers?

4 = a I would have thought?

That wouldn't make much of a hustle. I just noticed that wizard of odds have a table on this one:
♠ Ask the Wizard! No. 160 (April 5, 2006)

5 = 50% still, but again I expect to be wrong as its too obvious.

question 5 bothers me too but there you go, I asked it now
Math Forum - Ask Dr. Math

[Andy]

Then the test would have been completely useless rather than just slightly useless.

Im sticking to my answer, the first line is "It is known that in the general population a certain drug is taken my 0.5% of people" and the final Question is "what are the chances he is a user".

Well its 0.5% then as he is one of the general population and that first line is a fact, whether he works in your office, drives a big car, smokes, tested positive to that test and negative to an HIV test, and any other facts you would like to throw in wont change a thing.

The first premise is 0.5% of the general population take the drug,
He is one of the general population,
There is a 0.5% he takes the drug then.

Its a tautology, its true by definition. It may not be mathematically true or the point you are trying to make, but it is still true given the structure of the question.

A triangle has three sides
This shape has three sides
This shape is a triangle

Im sticking with my answer (even though its come at it from a different angle)