Basically what 2FA said.
To maybe add an example to make it more clear regarding point two.
If you have a situation where an event has say a 1% chance of occurring say each unit of time(or event), and that event is based on a random set of conditions, then no, the chance doesn't increase if it hasn't happened the previous event.
For example, imagine a spinning wheel in a carnival, or roulette table(same idea), then each time that wheel is spun, every chance is effectively reset. Just because say it didn't land on red 22 last spin, then the next spin has exactly the same chance of landing on red 22 as it did on the last spin. Same as with that carnival wheel. This is a pretty common misconception, that it changes, many people assume it must change the odds, it doesn't.
However if you have a system where the probability is based on a set of events which are not random. Like say there is a 0.0001% chance of your car tire blowing while driving 1km. Then the next 1km, if you keep driving that car over time, the chance of that tire blowing will change because there is a sort of wear that happens that changes the conditions for the next time. Another example is say the chance of an earthquake each year. Because with each year the pressure being build up increases if that pressure is not released. So in that case, every year that stored energy along that fault line is not released by a seismic event, then the next year the chance increases.
So people think that because their numbers in a lottery haven't come up yet, that means there is more chance it will. The way the lotto machines work, that is not true. Same with so many areas people think like that. Also with computers, most randomness on a computer is based on a combination of some state the computer can read in that particular moment(reading the exact clock tick at that moment it is called, or the last keystroke, or better sometimes is the moment some radioactive element decays) and then using that number in a algorithm that relies on ideas from cryptography(primes, keys, etc) to generate a new value from that 'random event' that theoretically makes it nearly impossible to deduce the original random event's value. Or even better relying on an algorithm that uses several 'random' states together. Hackers used to be able to exploit 'pseudo-random' number generators in old machine operating systems, but random number generators in programming has since come a very long way.
Regarding the first question, it depends again on the relationship between those probabilities. If they are again both completely random, then it is just a matter of multiplying the probabilities against each other. So say one event 1A has a 10% chance of occurring(90% of not occurring (we can call 1B)) and in a different unrelated system, another event(2A has a 20% chance of occurring(and 80% chance of not occurring 2B). Then the chances would play out like...
Chance of 1A and 2A is 0.1 * 0.2 = 0.02 = 2 %
Chance of 1A and 2B is 0.1 * 0.8 = 0.08 = 8 %
Chance of 1B and 2A is 0.9 * 0.2 = 0.18 = 18%
Chance of 1B and 2B is 0.9 * 0.8 = 0.72 = 72%
Total = 100%
Again as 2FA said, that all changes if there is some other relationship between those events you are speaking of. For instance, if that propability represent the chance of two different people winning in a fantasy pool. Person A has 90% chance, and person B has 80%, then if you know that say person say didn't win, that does have an effect on the probability of person B willing, since his chances can now go up or down. If you knew nothing of their 'picks' then the default would be to assume that person B now has a better chance of winning since there are effectively less people in the pool now. But if you know more information, and know say that person B has many of the same picks as person A's team, then it is likely than that person B is also doing crappy in the fantasy pool, and so his chances will go down most likely.
Again so it depends very much on what the relationship is between those two events.
(Ok this is a post edit note, just ignore the rest unless you are actually also interested in deep learning concepts around probability as well...)
And incidentally, that is effectively what deep learning is all about, pulling out the probability of every possible combination of events you can feed into the matrix, a giant sparse matrix, and once you solve it, what happens is the computer is essentially combining all these probabilities together to find patterns so complex that humans would not be able to keep track of all the variables. Like face recognition, is largely about identifying the probability associated with each pixel in an image(a huge matrix of probabilities associated with colors) For instance in a real basic kind of sense, a person with dark black hair will have way more pixels having dark black in them then vs someone who has bright blue hair for instance. And so when you combine not just the color of each pixel, but the combination of colors beside each pixel on an image. You get a giant bloody matrix of probabilities, and out of that you can deduce what the probability is that the image is the same as some set of images of a person the computer build a probability matrix model of in a training stage. Basically, run through a bunch of images of a person, build the probabilities out of that around what color each pixel will be against all the other pixels in the image, and then compare that against a new image to deduce if it the same person.
In that case, in the case of deep learning, you assume you mostly don't have any clue what the underlying structures are, and rely purely on historic data say to form the best guess. And in doing so computers find out crazy relationships, like the computers that pick stock, they use any information they can find they think might have some correlation with stocks going up or down. This includes sometimes millions of internet pages, that the computer just might pull out some crazy correlation, that says that every time Trump comes on tv, and this particular blog writes an article about it, this other stock, that most people might think has no correlation between those events, might go up or down. And a human would ignore that because they can't understand the correlation, but a deep learning machine says, I don't really care why it happens, only that there is a strong probability between those events, and so it 'guesses' better effectively because it isnt' prejudice, and it also is using millions upon millions of small correlations that sometimes can all add up to a very strong ability to predict events...
Sorry I rambled endlessly there. Cleary I need to be doing something else.
I hope there was something in there that perhaps helped...