Weekly puzzle 8 – solution
Clever Lawyer Solution
The clever lawyer was able to deduce that the probability that the witness identified the murderer is not the same as the probability he identifies the colour correctly.
There are also two possibilities – the colour was green and correctly identified, or the colour was red and incorrectly identified.
The probability that a Green gang member committed the murder given that the witness identified green is
(0.8 X 0.85)/ (0.8*0.85+0.2*0.15) = 0.9577…
This exceeds the 95% minimum certainty required by law.
As it turns out, the witness needs to be 82.6% effective to achieve the conviction.
This puzzle has real life connotations. Next time you check the weather forecast and the forecaster says it will rain, and it’s dry, think of this problem as it explains why they are usually wrong. Given that the weather forecasters are 85% accurate and that is dry 95% of the time, it turns out the probability of it raining when they predict it is only 23%, whilst the probability of being dry if predicted is 99%.
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It took me a while to understand the answer to this one. My confusion lay in the fact that the witness was “convinced” it was a green gang member. I still should have figured it out, but the problem would have been much clearer if it had been stated that he identified the culprit as wearing green clothing or something to that effect.
Good puzzle, and an important concept for honing one’s critical thinking skills.
BTW, your google reader has gone a bit haywire – it is showing this solution (or maybe just the most recent post) under the last 10(ish) posts.
Just not to sound too obscure, if we continue the digits of the number, will have 0.9577464… and that’s from where it comes my 7464…
I keep being continuously surprised how much important is this puzzle. I knew the reasoning about the weather, but forgot to relate it with the puzzle. About bad news from doctors: it’s justified to have hope when some exam points us a bad illness. If the normal state is to be healthy and the percentage of sick population in that particular disease is very low against the rest, then the false positives indicted by the medical exam might be comparable with the true ones, and offer the hope the exam is wrong even if very accurate. Of course, when it matters, there are ways to reduce this kind of uncertainty: There is nothing to say that the exam’s accuracy must be symmetrical regarding false positives and false negatives. Also, it is not all the population who usually submits to medical exams but the ones who felt ill (and the percentage of sick between those might be indeed bigger than in relation to the general population). That raises interesting thoughts about public obligatory campaigns to detect diseases versus voluntary ones, panic versus calm.