What are the chances of anyone accidentally creating the same Bitcoin address?
In theory, that's possible. In practice, that's impossible.
Take note that there are 2
160 valid bitcoin addresses (There are even more addresses if consider all types of bitcoin addresses).
Is it remotely possible or there is some kind of checks before network, software, God or something would do a check first before approving the Bitcoin address.
No.
Note that your bitcoin address in generated through some mathematical calculations from a random number and you don't need internet connection for that.
Well very good question raised by OP (curious one) and nicely explained here as to what can happen. Good to know that we will tiny to no possibility of having same address getting generated.
There are hundreds of examples that explain how it is not possible or highly unlikely to get messed up in this situation. I think if it happens it would start the chain reaction of chaos on a blockchain. Imagine spending bitcoins twice because it was initiated from the same sender address. Not sure whether two owners will get benefited if someone credits to that address (off course in parallel universe if the addresses are same!)
But then, again I got impressed with the following example from one of the article associated with likelihood of hitting same person when you walk around FRANCE. Very nicely explained, everyone must read this to understand its not possible that we will have same address again.
What does it match to?
Let’s try to evaluate the size of such a figure through an analogy. Imagine that you have a computer capable of generating 400,000 Bitcoin addresses per second, which already requires significant computing power.
Start the computer, and go for a walk. For example, complete the Tour of France (6397 km), walking at a normal pace of 4 km/h. Once you have finished your walk, say hello to a random person, and start your walk again. Once you have said hello to 66 million inhabitants, and completed the Tour de France each time, visit one of the 36,000 French municipalities.
Now repeat your tours of France, your greetings to each inhabitant, and when you have finished visiting all of the French municipalities, you will probably have generated a Bitcoin address collision with your computer.
The computer in question would then need 10 million billion of hard drives to store such a large amount of data. Unfortunately, here is the bad news: there is no guarantee that the address on which you generated a collision contains bitcoins: at present, only 10 million addresses have been used on the network.
The probability of generating a collision on an address containing bitcoins is therefore much lower, and it is highly likely that the Earth would have disappeared well before you finish your tours of France.
Conclusion: Don’t worry
All this is simply to say that one should not worry about the possibility that an address collision is generated by two different wallets. The creator of the Bitcoin network has devised a particularly well-adapted system, which, through decentralization, ensures that everyone has the opportunity to use the network without having to register the addresses of their wallets with anyone, and this without fear that two people will ever use the same address.
Full read here. The article also explains numerical calculations behind the possibility of this event happening. How there is 99.999% of chance it wont happen at all. Just a quick 2 minutes read.
What if my wallet generated an existing Bitcoin address?