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A Math/probability Question


Wayne-1

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there is a disease that afflicts 1 out of every 1000 people...

scientists develop a test that:

- if you have the disease, will identify it 100% of the time...

- if you don't have the disease, will be accurate 95% of the time and provide a 'false positive', i.e. it says you have the disease when you don't, the other 5% of the time...

if a person takes the test and it says the person has the disease, what are the chances the person actually has the disease?...

if you want the mulitple choice options, look below the pic... the real brainiacs will get it without the choices... ;)

I'll post the answer if I remember, PM me if I forget and you want the answer...

according to the BBC, this question (with the multiple choice options) was asked of 60 academics and senior medical students at Harvard Medical School during the 1970s and only 18% of them got the correct answer...

funny-pictures-cat-says-your-disease-is-incurable.jpg

a ) 95%

b ) 100%

c ) 2%

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Question doesn't make sense, because that 1 out of 1000 thing is irrelevant and none of the possible answers add up. If the question was

"If you take the test, what is the chance you have the disease" then it would make more sense.

don't blame the question if you can't get the answer...
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Question doesn't make sense, because that 1 out of 1000 thing is irrelevant and none of the possible answers add up. If the question was

"If you take the test, what is the chance you have the disease" then it would make more sense.

Think again. ;)

I'd post the correct method of thinking for this, but I want to give people time if they're still thinking so they don't accidentally read my post. If Wayne hasn't posted it by tomorrow at some point in the evening I'll go ahead and do it.

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Without looking at multiple choice or any replies:

For 1000 people, 1 will have disease and have it correctly identified, and there will be 999*.05 = 49.95 people given false positives.

So the chances are 1/49.95 or roughly 2% that the person actually has the disease and 98% that they are in the false positive group.

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Question doesn't make sense, because that 1 out of 1000 thing is irrelevant and none of the possible answers add up. If the question was

It makes all the difference in the world, think of this example: if 1000 of 10000 people have the disease, the answer would be 100%, which it clearly isn't in this case.

"If you take the test, what is the chance you have the disease" then it would make more sense.

Well there wouldn't be much point to that question since the first line gives you the answer. The chance you have the disease is 1 in 1000, it doesn't matter if you take the test or not (assuming you aren't more likely to get tested if you have the disease or anything).

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It makes all the difference in the world, think of this example: if 1000 of 10000 people have the disease, the answer would be 100%, which it clearly isn't in this case.

Well there wouldn't be much point to that question since the first line gives you the answer. The chance you have the disease is 1 in 1000, it doesn't matter if you take the test or not (assuming you aren't more likely to get tested if you have the disease or anything).

Maybe I'm reading the question wrong, but the actual question being asked says if the person takes the test and the test deems them positive, whats the chances they have the disease. So if the disease affects 1 out of every 1000, why do those extra 999 people have any relevance if the actual proposed question does not involve any of them?

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Maybe I'm reading the question wrong, but the actual question being asked says if the person takes the test and the test deems them positive, whats the chances they have the disease. So if the disease affects 1 out of every 1000, why do those extra 999 people have any relevance if the actual proposed question does not involve any of them?

It doesn't directly involve the other people, what's important is the chance that the person actually has it is 1/1000.

Because if the person is told they have the disease, either they have it correctly identified, or they have a false positive. To get the proportion of one to the other you need to know the chance they actually have it

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It doesn't directly involve the other people, what's important is the chance that the person actually has it is 1/1000.

Because if the person is told they have the disease, either they have it correctly identified, or they have a false positive. To get the proportion of one to the other you need to know the chance they actually have it

I understand that, but the exact question is,

what are the chances the person actually has the disease?...

Upon being told they have had a positive test, which means we've already identified they were the 1 out of 1000, which again leads me back to why did we need to know that in the first place.

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okay, it seems almost everyone got it...

they didn’t say on the show, but I’m guessing the Harvard test was less a test of math ability but rather a psychological test to show how a seemingly ‘accurate’ test, correct 100% of the time for positives and 95% of the time for negatives, really isn’t accurate unless positive test results are double-checked -- doctors certainly better re-run the test before telling a patient he/she has a serious illness... they said over half of the respondants at Harvard picked 95%... I think if they changed the obviously incorrect 100% choice to something like 5% or even 50%, even fewer of the respondants would have gotten it correct...

another question to follow...

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A country is preparing for a possible future war. Its tradition is to send only men into battle, so they want to increase the proportion of males to females in the population.

A law is passed which requires every married couple to have children and to continue to do so until they have a boy.

What effect do you expect this law to have on the population?

a ) More (by proportion) boys will be born.

b ) More (by proportion) girls will be born.

c ) There will be no change to the gender balance.

two important assumptions:

1) the chance of a baby being a boy (or girl) is exactly 50-50...

2) once a boy is born, that couple stops having children...

(this is the scenario on the BBC website/show, it's pretty stupid, so just focus on the math part of the problem)

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A country is preparing for a possible future war. Its tradition is to send only men into battle, so they want to increase the proportion of males to females in the population.

A law is passed which requires every married couple to have children and to continue to do so until they have a boy.

What effect do you expect this law to have on the population?

a ) More (by proportion) boys will be born.

b ) More (by proportion) girls will be born.

c ) There will be no change to the gender balance.

two important assumptions:

1) the chance of a baby being a boy (or girl) is exactly 50-50...

2) once a boy is born, that couple stops having children...

(this is the scenario on the BBC website/show, it's pretty stupid, so just focus on the math part of the problem)

C: no change

Couples having a first child, 50% have boys, 50% girls.

Those who have a girl go on to second child, 50% have boys, 50% have girls

Those who have a girl go on to third child, 50% have boys, 50% have girls

and so on to infinite.

This is a trick question because when a couple stops having children doesn't have any effect on the answer.

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Just checking the math:

A town with 40 couples

20 boys 20 girls

10 boys 10 girls

5 boys 5 girls

5 girls: 2 boys 2 girls if it's a boy, then there is 38 boys and 37 girls. If it's a girl then there are 37 boys and 38 girls, and the number of girls increases until a boy is born. It could be both depending on the situation. Man, maybe I'm overthinking it... I'll go with C because of my uncertainty.

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Man, maybe I'm overthinking it...

The way you looked at it actually makes me think that boys are slightly more likely.

Let's say you have 32 couples, assuming 50/50 chances, at some point you get down to 1 couple left and boys and girls born are equal

At this point, there are 3 possibilities:

1.) Last couple has a boy, so 1 extra boy overall (50% chance)

2.) They have a another girl (50% chance), which is either followed by

2b.) another boy, ending them at even (50% of #2, 25% overall); or

2c.) another girl, which will inevitably end with more girls than boys (50% of #2, 25% overall)

The interesting thing is that #1 is the case of more boys, but #2 includes a 50% chance of more girls and 50% chance of even number, so if you break it down that means:

50% chance: there is 1 more boy

25% chance: boys and girls are even

25% chance: there are 1 or more extra girls

So statistically, you are more likely to have more boys (well 1 more boy to be exact).

However, I assume the question is assuming infinite couples, and not getting wrapped up in the small details, so I think the correct answer is even.

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So statistically, you are more likely to have more boys (well 1 more boy to be exact).

However, I assume the question is assuming infinite couples, and not getting wrapped up in the small details, so I think the correct answer is even.

you've been pretty much right on the mark: your answer, your explanation, the assumption that there aren't a small, finite number of couples... that 'last' child should not change the 'proportion'...

btw, these are all the answers that were given on the show/website so don't blame me if you think the answers are wrong ;)

and I'm saving the one that will probably cause the biggest argument for last :P

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this is one I'm sure many of you have already heard of...

for some reason the person creating the problem picked these items: a wolf, a goat, and a basket of cabbages for a farmer to be taking to market... he has to cross a river in a small boat in which he can carry only one of the items at a time... the problem is if he leaves the wolf with the goat, or the goat with the cabbages, one will eat the other...

how does he make it across the river? (the BBC question just asked how many trips, but if you know how he does it, the number of crossings is also known)...

since that question is easy/common enough that many of you will probably know the answer immediately, here's a 'bonus' one for discussion:

A magician pulls out an ordinary deck of playing cards and gives it to a member of the audience to shuffle.

The deck of cards is then put inside an empty box.

The magician asks another audience member to name a card. They say, "Ace of Hearts".

What is the probability that the magician then pulls the Ace of Hearts out of the box?

a ) unknowable

b ) 1 in 52

c ) 100%

a hint:

(to be fair, I have to warn you that unlike the other questions, you need to consider every element of this question)

three more questions to go: one is just trivia, another will require real math/probabilty knowledge since they don't explain how they got the answer (and it's been so long for me that I got it wrong :blink: ) and one that is still argued about even among statisticians/mathematicians...

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