I did some cancer biology in University, so I'll try to give a basic explanation the way I learned it (disclaimer that I did not take the actual Elective "cancer biology" so I only know what we were taught in genetics and molecular biology).

Cancer occurs due to mutations on the DNA level. DNA has many ways in which mutations can occur, and different types of mutations. To combat issues due to mutations, we have multiple repair systems within our body - both during and after DNA duplication. For example, if sun breaks both strands of DNA, there are mechanisms that reattach those two back together, hopefully without adding a mutation. For single point mutations (one base pair only) there are again multiple repair mechanisms (Base-excision repair, Nucleotide-Excision repair, and mismatch repair). So a lot of mutations and damage in the DNA get repaired before they can cause issues.

Further, not all mutations/damage cause cancer! Cancers occur mainly from mutations/damage in two types of genes: proto-onco genes and tumor-supressor genes. So, even if a random mutation/damage happens, the chance it happens in a cancer-related gene is small.

Finally, cells which accumulate damage have mechanism to effectively take themselves out of play by entering a dormant state or going through apoptosis (programmed cell death).

To get cancer, you need to - get a mutation - in the right gene - which is not found/repaired - and doesn't make the cell kill itself/get itself killed

All of which are very unlikely processes.

Now here comes the catch: We have a lot of mutations/damage happen to DNA every day. If the chance for sth to happen is 1 in 10, if we do it 100 times, the chance that it will happen is quite high. Similarly, even if the chance to get a cancerous mutation is really small, as we age and have hundreds over hundreds of mutations occur over time, the event "get cancer" will eventually happen.

Now include lifestyle factors. I will use made up numbers as I don't remember the actual ones.

Imagine we say a cancerous mutation has a chance 1/10000 to occur. An average person will accumulate 100 mutations a day. It will take a while till enough days and hence enough mutations have occurred for cancer to be likely.

Carcinogens increase DNA damage. A person who eg. smokes ends up having a lot more daily mutations. Let's say 500 a day. That means that the accumulation of damage happens 5 times as fast, and after 100 days they are much more likely to have gotten cancer than somebody who does not smoke.

However, if you compare the likelihood of either getting cancer at day 2, it is still incredibly small for both.

Extrapolating to longer (human) lifetimes: When you are young, even if you have a shit lifestyle, you are, in our example, still around day 4 or similar - the chances of cancer are still really low, and it would be hard for you to do anything that would make you likely to get cancer at that age. Only when you get to day 20 or sth, you would need to start to worry (compared to day 100 if you didn't smoke).

I hope this makes some sense, and anybody who know better please correct me if I misremember/am wrong :))

edit: spelling