The Big Risk Test - Probability Question

Discussion in 'Mathematics' started by harderfaster, Apr 28, 2011.

1. harderfasterNew commenter

"A criminal hides in a room with 99 innocent people. You have a lie detector that correctly classifies 95% of people. You pick someone at random, wire them up to the machine, and ask them if they are the criminal. They say 'no', but the machine goes ?ping? and says the person is lying. What is the chance that you have caught the criminal?"

The possible answers are: 17%, 5%, 83%, 50%, 95%.

Source: https://www.bbc.co.uk/labuk/articles/risk/ - I think it's in section 8.

2. Betamale

Hi
Are you asking for help on the question?

None of the above answers is correct!

P(ping) = 0.99 x 0.05 + 0.01 x 0.95 = 0.059

P(ping and criminal) = 0.01 x0.95 = 0.0095

P(criminal | ping) = 0.0095 / 0.095 = 0.161

So the answer is closer to 16%

4. sapphire14New commenter

The question stated "You have a lie detector that correctly classifies 95% of people." so if the person answered 'no' there is a 95% chance that they are telling the truth, and a 5% chance they are lying ie are the criminal, so answer is 5% - the rest of the question is just fluff to me!

5. harderfasterNew commenter

No, I was hoping to spark debate, given that my answer was not any of the options (but has been given here already)...

6. pipipiNew commenter

My way of thinking gives 4 outcomes (without denominator of 2000)
Ping and innocent =0.05*0.99 =99
Ping and criminal = 0.95*0.01 = 19
Noping and innocent = 0.95 * 0.99 = 1881
Noping and criminal = 0.05*0.01 = 1
(the values all sum to 2000).

So if it pings ( which can happen in 99+19 cases) then the probability is 19/118 = 0.161....
I'm not sure if I should consider the case where it doesn't ping but it is still a criminal?

sapphire14, I really hope you don't teach probability. Get a book out and have a look at conditional probability. This type of question is very standard. In fact, there was a very similar one in New Scientist a few weeks ago.

Incidentally, this particular scenario, and there have been plenty others like it, illustrates the dangers of the jury system. The majority of jurors are relatively innumerate, and therefore totally incapable of correctly weighing up this kind of evidence - frightening!

9. ResourceFinder

One of the problems with this question is that a lie detector does not give 95% in both directions

So the chance of it mistakenly identifying a guilty person as innocent is not the same as a the chance it will identify an innocent person as guilty

The problem states that the lie detector correctly classifies 95% of people. Therefore, if I had 100 truthful people it would not ping for 95 of them. And if I had 100 liars it would ping for 95 of them. My earlier calculation follows, from this reasoning.

11. siddons_sara

I'm 100% with David on this one. Not a particularly interesting question, IMO.

12. ResourceFinder

Clearly

I was responding more to
a) the validity of your question and
b) your concern that jurors would find this an issue

13. AnonymousNew commenter

Used to have something similar in my old life doing blood tests.
A blood test is 95% sensitive. In otherwards it detects 95% of all cases.
It is 95% specific - in otherwards, 5% of the positives are actually false positives.
You test positive. Have you got the disease?

Unfortunately, the vast majority of jurors, on being told that a lie detector was 95% reliable WOULD assume that a ping meant that they had got caught the villain with 95% certainty.

There's been quite a bit of stuff on this kind of thing. I can't remember the details, or where I saw it, but there was one scenario where two separate pieces of evidence, that each by themselves indicated guilt, when combined LOWERED the likelihood. And, while DNA is a wonderful forensic tool, often the probabilities quoted are not correct.

15. florapost

in cot death cases, the problem with the stats quoted, i think, was that cot death and any causes were assumed to be random, as tho god/nature flung random genes into each child at conception, rather than looking at natural frequencies
but why should juries understand stats? why don't defence councils call maths expert witnesses to explain?
i keep thinking i would like to adapt the lie detector problem (i also came across it as a medical test problem irl) for my year 6's as having an alien detector - how good would it be at finding if one of them was a martian - but am embarrassed to admit haven't got round to it yet

16. FlippantFlyer

The CAT Scores as Indicators of GCSE Results charts amuse me as an example of the distortion caused by the use of statistics:
http://www.gl-assessment.co.uk/education/resources/cat3/indicators.asp?css=1
A child scoring 100 to 104 in maths on CAT3 looks favourite for a C! . . . until you look at the probability scale on the left. In fact approx 57% are not likely to get a C. Further, its probable that around 25% won't even get at least a C. The accompanying table shows just how blurred the picture really is.

17. valed

! ?? Relevance??

18. FlippantFlyer

Try reading the post above that one. Do try and keep up!

My measure?