(First off, I'm not that great in Math. So bear with me)


Let's suppose that someone slashes your tires. Assuming that the damage gets fixed soon after, and it happens again within a certain period of time (2-3 weeks), what is the probability that that crime was committed by the same individual?

we need to consider a few other pieces of info..
1) nature of the crime (not extremely serious, easy to commit without being caught)
2) The fact that your tires have never been slashed previously, or not for a long time (say 4-5 years)
3) There is no way of telling what was used to slash the tires is the same or different, and no physical evidence (ie shoeprints, DNA) that anyone was ever there.
4) You have a pretty good idea who slashed the tires the first time, but do not know the probability that he/she slashed them in the first place.

Without worrying WHO slashed the tires, can you calculate the probability that it was done by the same person along with a standard deviation? I can intuitively say that it was done by the same person, but how do I prove >0.5 probability that it was?

-Thanks

I would probably try something to this effect. First you would use an average slash rate of 1 slash every 2-2.5 years (you might want to lengthen that since we could be observing at an unusual point in time). Then assume that the exact timing of when the next slash occurs is independent of the time that the previous slash occured. This would mean that the timing of the slashes follows the Poisson distribution ( http://en.wikipedia.org/wiki/Poisson_distribution ). You then calculate the likelyhood of one slash occuring 2-3 weeks after the previous slash given that particular Poisson distribution. If that is P less than .05 then you are 95% confident that the slashing is not random.

If it is not random you are out of the arena of probability and standard deviations. Then you can start talking about motives, knowledge, etc. Obviously in order for it to be non-random then the person who slashed the second time would have to know about the first slashing. So that would limit it to the original slasher plus anyone who knew about the first slashing (you, the repairman, any friends or family you told, friends and family that the first slasher may have told, etc.). Of those, obviously the motive is probably strongest for the original slasher, but again, since it is non-random you can't really talk about probabilities or standard-deviations.

-Dale

Thanks! Your reasoning makes sense. So it CAN be proven that it was likely the same slasher independant of evidence? I wonder how that would hold in court. (May take me a while longer to understand Poisson distribution)

I wouldn't say that. What can be proven is that the slashings are not random. That still doesn't mean that it was the same person. In other words, if the burden of proof is "beyond a reasonable doubt" then it would be reasonable to think that the second slasher could be a friend of the first slasher. The defense could even argue that you had motive to do the second slashing yourself in order to frame the first slasher.

All I know how to prove would be the non-randomness. Then non-randomness implies that the second slasher knew about the first slashing. From there you have to make a list of who knew about the first slashing. Everyone on that list is a suspect, including yourself. Then you have to look at evidence, motive, opportunity, alibis etc. You would have to establish to whatever legal criteria that the first slasher is the only one on the list with the motive etc.

-Dale