58% Proof By Induction, 32% Random Chance

Mathematicians use proof by induction all the time, and while considered a "forced" proof, it is still a valid way to prove something. The issue, however, is that this does not work in the real world, which is to say, just because someone does well on a test one week, does not mean that they will do well on the next test. But why?

Two possibilities emerge: either proof by induction is flawed, or something is not being accounted for when induction is used. Since the method is sound, (as it has continued to be used), then something is not being accounted for. And this does make sense.

In the example above, only one variable is accounted for: is there a test? In actually, however, there a millions of variables in play, some of which are more important than others (one being: did you study?). Because these variables are not taken into account, or ignored, then incorrect results are the result.

For example, you see a friend with lots of blue clothing, and whenever they receive a piece of blue clothing as a gift, they are genuinely happy. Using proof by induction, you infer that every time he/she gets blue clothing they will be happy. This is a simplification through a generalization, and while it makes your life easier, it does not lead to a guaranteed good gift giving method.

Now, it does increase the probability that your friend will like the gift. But, they could absolutely hate scarves, in which case a blue scarf would be a bad idea. Still, the chance that your gift will be successful does increase. Which brings me to the summation point:

Patterns do not prove anything, but instead give background for claims. There are always multiple models and theories which fit the same data, however the one which is most effective and simplest is repeatedly chosen, as science reflects. So do not think that proof by induction is solid, just think of it as a quick way to improve your chances of success.