# A conceptual probability doubt.

Assume that a store manufactures N guitars. Every guitar has a probability P, of being defective. If I buy a guitar, and it turns out to be defective, does it change the probability of another guitar being defective?

One way to see it is that every guitar is manufactured independently and hence the probability shouldn't change.

Another way to see it would be once everything is manufactured, there is a finite fixed number of guitars that are defective (let's call it X, which is approximately PN). So finding one defective guitar implies the probability of finding another defective guitar becomes less : ((X-1)/(N-1)) instead of (X/N).

I am tempted to assume that each guitar is in a quantum state (mutually non-entangled, obviously) such that before we check a guitar, it is in a superposition of defective and non-defective. Which supports my argument but I am not able to convince my friend (or myself) why that should be the case in a classical system.

• If you are assuming each guitar is built according to a IID Bernoulli process as to whether it’s defective or not, then the probability does not change by definition—it is, as you say, an independent sequence of trials with constant chance of success (defective, here). Are you assuming this? If not, well you have some options. You could assume independent trials but not identically distributed, so each guitar has it’s own parameter $p_n$, chance of being defective but one guitar being defective is still independent of others. You could assume a Poisson process etc. So, what do you assume? – Nap D. Lover Sep 22 '18 at 0:13
• In addition to my above comment, do you really think it’s necessary to assume some sort of “quantum state” for such a problem that is already amenable to classical models aforementioned? – Nap D. Lover Sep 22 '18 at 0:15
• Finally, if you’re asking about, for a given luthier, the empirical proportion of defective guitars built and whether it changes over time—well that is an empirical statistics question that requires data that you can then test if an IID Bernoulli process is appropriate. Or, if you find the empirical proportion of defects is not constant over time, maybe you will use a non-homogeneous Poisson process. Perhaps defects increase during busy periods (holidays?). You need to see the data to pick a model that realistically describes the real-life phenomena. Hope this helps. – Nap D. Lover Sep 22 '18 at 0:17
• Thank you for the comments. I assumed the "Quantum model" only because I thought that the whole framework of superposition of orthogonal basis states perfectly describes what probability looks like in "real life". I couldn't find a logical distinction between a hypothetical guitar with a certain probability of being defective and a similar case about a spin of a subatomic particle. I 'm still not sure if it's a valid argument. – Indroneil Kanungo Sep 25 '18 at 22:59