Test of risk assessment
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Test of risk assessment
I came across this as a simple test of the ability of bankers to assess risk - indeed not just bankers, any typical human!
There is a fatal disease that kills 1 in 1000
A test has been developed that is 95% accurate.
Your doctor tells you that your test is positive.
What are your chances of surviving?
There is a fatal disease that kills 1 in 1000
A test has been developed that is 95% accurate.
Your doctor tells you that your test is positive.
What are your chances of surviving?
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Best start living like I am infected just in case, if it goes to court I can plead temporary insanity due to the stresses of the potential infection..If I die it'll be happy.
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If the disease kills everyone who catches it, and you're told you have a 95% chance of having it, then there's a 95% chance you'll die. The prevalence of the disease across the general population doesn't matter. But that's NOT what a "95% accurate" test means.
If you take the test and are healthy, there's still a 5% chance that the test will tell you that you have the disease - and that's much more likely then you actually having the disease.
There are four cases to consider:
a) you are healthy, and the test says so. p=0.999*0.95 = 0.94905
b) you are diseased, and the test says so. p=0.001*0.95 = 0.00095
c) you are healthy, but the test is wrong. p=0.999*0.05 = 0.04995
d) you are diseased, but the test is wrong. p=0.001*0.05 = 0.00005
(Sanity check: 0.94905 + 0.00095 + 0.04995 + 0.00005 = 1)
You're told that the test is positive, so you're in (b) or (c). You hope for c, of course, and the chance you're in the b group is 0.00095 / (0.00095+0.04995) = 0.01866, or about 1 in 54.
If you take the test and are healthy, there's still a 5% chance that the test will tell you that you have the disease - and that's much more likely then you actually having the disease.
There are four cases to consider:
a) you are healthy, and the test says so. p=0.999*0.95 = 0.94905
b) you are diseased, and the test says so. p=0.001*0.95 = 0.00095
c) you are healthy, but the test is wrong. p=0.999*0.05 = 0.04995
d) you are diseased, but the test is wrong. p=0.001*0.05 = 0.00005
(Sanity check: 0.94905 + 0.00095 + 0.04995 + 0.00005 = 1)
You're told that the test is positive, so you're in (b) or (c). You hope for c, of course, and the chance you're in the b group is 0.00095 / (0.00095+0.04995) = 0.01866, or about 1 in 54.
Last edited by AndyC_772; 07 January 2011 at 08:00 AM. Reason: you hope to be in the 'live' group, probably...
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I think you have done what the bankers and did and create the 'illusion of validity'.
If I read your answer correctly you have calculated the chances of surviving a positive test as 4.995%. Is this correct?
If I read your answer correctly you have calculated the chances of surviving a positive test as 4.995%. Is this correct?
#14
I use this test interviewing for software engineering positions. Lots of people don't think about the false positive case.
Another favourite - I give you a cube of cheese and ask you to cut it into 27 smaller cubes. What's the minimum number of cuts required? You are allowed to stack the pieces any way you like for each cut.
Another favourite - I give you a cube of cheese and ask you to cut it into 27 smaller cubes. What's the minimum number of cuts required? You are allowed to stack the pieces any way you like for each cut.
#15
His answer is right. The question illustrates how what seems like a statistically fairly accurate test (95%) can give a misleading impression when applied to rare events. Even though the test is positive, it's much more likely that the test is wrong and you are healthy.
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Anyway - what do you think the answer is?
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My 98% is based on:
1 / 1000 chance you'll die
50/1000 chance test is inaccurate
Therefore 49/50 chance you'll survive = 98% chance of survival following the positive test.
(Ignores the chance of a negative test result being incorrect, but it's small enough that I can't be arsed to factor it in.)
D
1 / 1000 chance you'll die
50/1000 chance test is inaccurate
Therefore 49/50 chance you'll survive = 98% chance of survival following the positive test.
(Ignores the chance of a negative test result being incorrect, but it's small enough that I can't be arsed to factor it in.)
D
#18
I use this test interviewing for software engineering positions. Lots of people don't think about the false positive case.
Another favourite - I give you a cube of cheese and ask you to cut it into 27 smaller cubes. What's the minimum number of cuts required? You are allowed to stack the pieces any way you like for each cut.
Another favourite - I give you a cube of cheese and ask you to cut it into 27 smaller cubes. What's the minimum number of cuts required? You are allowed to stack the pieces any way you like for each cut.
nah
6 - It's six innit
CRAP - It's not 6 -It's 4
I think
Last edited by jods; 06 January 2011 at 09:45 PM.
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I always hated statistics, so I'm quite prepared to be proved wrong - but I'd be interested to know where my reasoning has broken down.
The cheese answer is 6, but it's the justification that's tricky. I consider this:
When you've finished your cuts, however you make them, you can stack the 27 smaller cubes as a 3x3x3 cube. The central cube has no external faces, so its faces must all have been made by your knife. It has six of them, so that's the minimum number of cuts you can possibly have made, however you chose to rearrange the pieces in the interim.
Last edited by AndyC_772; 06 January 2011 at 09:46 PM.
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The exercise is used to demonstrate how the human mind is not well designed to assess risk. It seems like a simple question and once explained the answer is simple.
Let's see if there are any more answers
Let's see if there are any more answers
#21
- 95 of the 100 ill people will get a positive result and 5 will get a negative result.
- 94,905 of the 99,900 healthy people will get a negative result and 4995 will get a positive result.
So, of the 5090 people who got a positive result, 95 are ill and 4995 are healthy. So, if you got a positive result your chances of survival are 4995/5090 ~= 98%.
Last edited by scud8; 06 January 2011 at 09:49 PM.
#22
I've calculated it as approximately 1.87%. [edit: this is the chance you'll die; you're hoping to be in group c, which has a probability of 98.13%].
I always hated statistics, so I'm quite prepared to be proved wrong - but I'd be interested to know where my reasoning has broken down.
The cheese answer is 6, but it's the justification that's tricky. I consider this:
When you've finished your cuts, however you make them, you can stack the 27 smaller cubes as a 3x3x3 cube. The central cube has no external faces, so its faces must all have been made by your knife. It has six of them, so that's the minimum number of cuts you can possibly have made, however you chose to rearrange the pieces in the interim.
I always hated statistics, so I'm quite prepared to be proved wrong - but I'd be interested to know where my reasoning has broken down.
The cheese answer is 6, but it's the justification that's tricky. I consider this:
When you've finished your cuts, however you make them, you can stack the 27 smaller cubes as a 3x3x3 cube. The central cube has no external faces, so its faces must all have been made by your knife. It has six of them, so that's the minimum number of cuts you can possibly have made, however you chose to rearrange the pieces in the interim.
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#25
I've calculated it as approximately 1.87%. [edit: this is the chance you'll die; you're hoping to be in group c, which has a probability of 98.13%].
I always hated statistics, so I'm quite prepared to be proved wrong - but I'd be interested to know where my reasoning has broken down.
The cheese answer is 6, but it's the justification that's tricky. I consider this:
When you've finished your cuts, however you make them, you can stack the 27 smaller cubes as a 3x3x3 cube. The central cube has no external faces, so its faces must all have been made by your knife. It has six of them, so that's the minimum number of cuts you can possibly have made, however you chose to rearrange the pieces in the interim.
I always hated statistics, so I'm quite prepared to be proved wrong - but I'd be interested to know where my reasoning has broken down.
The cheese answer is 6, but it's the justification that's tricky. I consider this:
When you've finished your cuts, however you make them, you can stack the 27 smaller cubes as a 3x3x3 cube. The central cube has no external faces, so its faces must all have been made by your knife. It has six of them, so that's the minimum number of cuts you can possibly have made, however you chose to rearrange the pieces in the interim.
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Most people give 5% as the initial answer.
The answer is actually 98% (this does not allow for a false negative).
It is a simple example but many houses of cards in banking were built on the 5% assumption - not the 98% reality!
The answer is actually 98% (this does not allow for a false negative).
It is a simple example but many houses of cards in banking were built on the 5% assumption - not the 98% reality!
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I guess that you would call it cheating though
You would have to use a molecularly fine cheese-wire and cut down one third in and parallel to one face.
Then, as the wire got one molecule from the bottom face, move it horizontally in a further third and then pull it up cutting back to the top (in a squared U-shape).
Thus you have almost achieved the initial two cuts used in the "standard" six cuts method - bar a very thin slice/layer at the bottom.
Repeat this action (twice) in a perpendicularly way each time to divide it up into almost 27 sub-cubes.
Then wait a few seconds for the mono-molecular layers to evaporate (or something) and "hey presto" - you did it in three cuts.
mb
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How about this one.
You have a pair of old fashioned scales.
You also have a concrete lump weighing 40lbs.
You can break the lump into four pieces.
How heavy does each piece need to be to enable you to use the scales to weigh any weight from 1lb to 40lb (in 1lb increments).
You have a pair of old fashioned scales.
You also have a concrete lump weighing 40lbs.
You can break the lump into four pieces.
How heavy does each piece need to be to enable you to use the scales to weigh any weight from 1lb to 40lb (in 1lb increments).