Does the day of the week affect the probability of picking a red ball out next? Does the weather affect it? Why not? Because those don't change the scenario: you're still drawing out of a bag of 37 balls. Now, ten times, you draw a ball one at a time and put it back. Is your bag now, after...
Photons don't experience time. It's the tradeoff for traveling at the speed of light. Therefore a "photon's standpoint" is meaningless.
Also, it is not true that light travels any distance in 0 time. In a vacuum, light travels 299792459 meters in one second.
Many of your questions are very...
In English:
A function turns one number, usually written "x", into another number, usually written "y". For example, y = x is a function. This turns the number x into a number y -- in this case, the number y takes the same value as x. Or the function y=4x turns a number x into a number y which...
See the "collectors" problem.
Basically, the problem is that if you're a toy collector, and you want to collect all possible 10 Burger King toys, on average how many times do you have to eat at Burger King? What is the probability, after n runs, that only m toys are missing?
You're collecting...
Exactly -- and that's why leaving out the dx makes no sense. That's what I said. The point is that the lack of a dx doesn't only mean that you don't know with what to integrate with respect to. Even if it was, it could just be implied for simpler integrations, like in the f-prime notation for...
To say that dx is simply telling us the variable to be integrated makes no sense. If it's just to tell you which variable to integrate with respect to, why use such a distinctive notation? For example, the variable could be written in square brackets at the top of the integral sign.
Correct me...
In my opinion, the confusion here is in the notation S(N). S(N) isn't a typical function. S(N) is in fact more like a statement. But an example will illustrate the idea much better than vague statements like that.
In our example, S(N) means "The number N has the property that if you sum up the...
In more casual terms, induction is taking a finite number of observations and "inducing" a larger conclusion. Proof by induction is called "proof by induction" because it requires only two observations: that it is true for at least one number, and then if it's true for that one number, then it...
Feynman mentions it in some of his literature, is this where you heard it? He read about the idea in Frederick Wood's Advanced Calculus, a now hard-to-obtain book, but a reasonably-sized library may have a copy. It does have a very good explanation of differentiation under the integral sign. If...
I recommend Richard Feynman's book, Six Not-So-Easy Pieces, or if you're truly inspired, the Feynman Lectures on Physics Volume I. The Lectures are very widely recognized as the book against which other physics books are measured, and Six Not-So-Easy Pieces is a subset of those lectures covering...