Relatively, there are a very Very VERY small number of configurations that result in there NOT being enough oxygen near your face holes to keep you alive.
And yet there are (e.g. industrial) situations where exactly that can happen!
Which is why I specified in my example:
"a typical room"and
"oxygen is RANDOMLY distributed throughout the room. It is RANDOMLY bouncing off the walls, ceiling, other air molecules, and ourselves"and I stated that I was talking about a situation where:
"EVERY POSSIBLE configuration of oxygen molecules in the room are EQUALLY possible"You are talking about a non-typical room, where the distribution of the oxygen is no longer RANDOM (as it is being influenced directly by some industrial process) and most importantly you are talking about a situation where some configurations of oxygen molecules in the room are either impossible, or at least less likely than others.
An analogous situation is when a random number generator isn't sufficiently random - the crypto seeds it generates may be a small population.
Correct. Private keys that are randomly chosen from the entire set of valid private keys are secure. Private keys that are randomly chosen from a small definable subset of valid private keys can be VERY insecure.
The lack of randomness may have insidious causes, for example seeding the RNG with the microsecond timestamp of a keypress. But it may turn out that USB is polled at a rate derived from the same microsecond clock. You may think the timestamp has 10 bits of entropy, but it really only has 3. This may seem to generate unique seeds, but alas, they're not good enough.
Which is why it is generally a bad idea to try to reinvent cryptography on your own without a SIGNIFICANT amount of education, study, and expertise. When it comes to maths and physics, what "feels intuitive" to someone that has a limited knowledge can often be COMPLETELY WRONG.
Slightly off topic for this thread, but related to our discussion about why it can be important to know the actual correct calculations:
Here's an example I recently heard about of how an intuitive feel for something that we ALL think we have good general knowledge of can be quite wrong...
Read the following and just think about this intuitively. This is not a "trick question", the answer just isn't as intuitive as most people might think it would be.
Assume a two lane road.
Assume two identical vehicles both traveling on the road in the same direction (one vehicle in each lane).
Assume that one vehicle is traveling at 70 miles per hour.
Assume the other vehicle is traveling at 100 miles per hour.
Assume that at the exact moment that the two vehicles are exactly next to each other, BOTH drivers simultaneously notice an obstruction up ahead that entirely blocks the road perpendicular to their direction of travel (in other words they both have exactly the same distance to travel at that moment before they will hit the obstruction).
Both drivers simultaneously hit their brakes and both apply the exact same maximum stopping force available to vehicle. Both vehicles are on identical surfaces and continue in a straight line towards the obstruction without any rotation of the vehicle.
The driver that was traveling at 70 miles per hour manages to stop within a fraction of a millimeter of colliding with the obstruction.
Which of the following is an accurate description of what happens with the vehicle that was traveling at 100 miles per hour:
The faster vehicle also stops before it collides with the obstruction
The faster vehicle collides with the obstruction and is traveling at a speed between 0 and 20 MPH when it collides.
The faster vehicle collides with the obstruction and is traveling at a speed between 20 and 40 MPH when it collides.
The faster vehicle collides with the obstruction and is traveling at a speed between 40 and 70 MPH when it collides.
The faster vehicle collides with the obstruction and is traveling at a speed between 70 and 100 MPH when it collides.
The faster vehicle collides with the obstruction and is traveling at a speed greater than 100 MPH when it collides.
Answer (copy/paste the following text or simply hit the "quote" button to read):
Intuitively, most people that are very familiar with general concepts of driving and stopping a vehicle could easily rule out the first and last options. However, intuition from common experience will generally lead someone to believe that the vehicles will both lose 70 MPH of speed, resulting in the faster vehicle colliding at 30 MPH. Some might realize that the faster vehicle has less time available to stop (since it is covering ground faster) and so they might guess between 40 and 70. Surprisingly, the correct answer is that the faster vehicle is traveling between 70 and 100 MPH when it collides (approximately 71 MPH)I was right by intuition.