Suppose we have a 1% chance that something happens.

The chance it doesn't happen the first time is 99%.

The chance it doesn't happen the second time is 99%, independently of the first (we do need some degree of independence here, or your statement isn't true). So the chance it didn't happen either of the first two times is 99% times 99%, or 0.99 times 0.99, or 0.9801.

The chance it doesn't happen the third time, still 99%. So the chance it didn't happen any of the first three times is now 0.99 times 0.99 times 0.99.

You can hopefully see a pattern. The chance for the event to fail every one of the first n times is 0.99 times 0.99 times ... times 0.99, n times. In other words, that chance is 0.99^(n). And the limit of 0.99^n as n goes to infinity is 0, meaning that the probability that you manage to dodge your low probability outcome gets closer and closer to zero the larger the number of attempts you have.

Nothing about this depended on the exact 1% / 99% probabilities. All we needed was for that 99% to be less than 100%, i.e., for the 1% to be greater than 0%. As long as that's true, given some non-zero probability of success p, (1-p)^n always goes to 0 as n -> infinity (and therefore the chance you fail every time goes to zero as well).

This guy gets it. Infinite series is a powerful tool.

Good answer but not really an ELI5 answer.

I wish someone can eli5 this answer, English isn't my language

Lets just be happy it's not a serie owned by Netflix other wise they would have cancelled it already

An event with only a 1 percent chance of occurring is not guaranteed to eventually occur. But empirically, an event with a 1 percent probability will occur an average of approximately 1 time in every 100 tests.

So if you run 1000 tests, empirically, we would expect to see 10 occurrences. Experimentally, this turns out to be the case. Even though results for each set may vary above and below what the probability predicts should occur, as the number of sets increases, the average of all the sets converges on the probability.

Imagine you have a dice with 1 million sides. Your chance of rolling 123,456 is 1 in 1 million, or 0.0001%. You start rolling the dice once a day. Each day the chance the dice will land on 123,456 is 1 in 1 million. That doesn't change day to day. Now lets fast forward in time 10 million days. You've rolled the dice 10 million times. Each individual roll was still a super tiny chance. But the odds that you won't have landed on 123,456 in those 10 million roles is approximately 0.0045%. (0.999999^10,000,000)

There is still a chance that you won't have rolled a 123,456 but it is very small.

This isn't actually strictly true. There's a tricky bit of math involved here, the idea of "certainty". If something is "certain" then we know that it will definitely happen. If something is "not certain" then there is a chance it won't happen, even if that chance is very small. If something is "almost certain" then we know that something will happen if we try infinitely many times.

If I flip a coin once, then I am not certain if it will land on heads at least once.

If I flip a coin a very very large number of times (like a billion) I am still not certain that it will land on heads at least once. That is because there is still a chance that I flip all-tails.

I I flip a coin an infinite number of times then I am almost certain that it will land on heads at least once. This is because in an infinite number of coin flips, the chance of all-tails becomes zero. We are "almost certain" that we will eventually flip a heads, but we are not "certain".

But here's the catch: it is not possible in reality to flip a coin infinitely many times, therefore in reality there is no way for a 100% chance of all-tails to happen. It can get very very very close to 100% but it will never be 100%.

It’s incredible I think I just posted a very similar question to this without having actually seen this until after I posted. What are the odds of that 😂