Some games don't have you randomly generate your hit points. Some games just give you a set number of wounds, or your wounds are based on other stats that you generate and can raise. However, quite a few of the 'really big' games out there - D&D, Pathfinder, Starfinder, etc - have you roll for Hit Points, and that can make for some interesting happenings in game. Today I want to talk about that.
Bad Luck Rolls
The basic problem you can run into with rolled for Hit Points is that you can end up with unusually low numbers for classes that are supposed to have a lot of HP. Sure, a Barbarian rolls a D12 per level, and a Fighter rolls a D10 per level while a mage only rolls a D4 or a D6 depending on the game, but if the Barbarian and Fighter keep rolling low you can end up in a situation where a character that is supposed to have a lot of HP has very few, which is bad when you also consider those classes are often at the most risk for taking damage as well.
Taking The Average
To counter this, some games suggest that instead of rolling for HP you just take the average. The average is whatever the total number of sides the die has, divided in two and plus point 5. So the average roll for a D10 is 5.5. The average roll for a D6 is 3.5. You get the idea.
This is where most adventures and monsters are balanced against - against an average number of hit points. This also gives some stability to things, as it means your Barbarian will always get 6.5 HP per level, and thus should never be in a situation (barring some weird constitution scores) where they are lower hit points than a mage or a rogue.
Roll or Take The Average
A different fix I've seen more of lately is having players roll, but if they get below average they take the average. Basically this means you get a shot at getting that sweet +12 hit points on your barbarian, but if you roll below 6.5 you consider yourself to have rolled a 6 and go from there.
I find I like this way because it lets the player try for a high result, but protects the game - and the character - from someone being significantly less sturdy and tanky than they should be. However, it does have a cost of itself.
Higher Than Average Hit Poitns
Whether from a string of good luck, or just what is mathematically likely to happen when low rolls are mitigated by the average score and high rolls are not, you can end up with characters with significantly higher HP totals than the average progression would expect.
This can lead to the "problem" of the PCs being more powerful than the game is balanced for. Able to take more hits than otherwise expected and thus able to do more in an 'adventuring day' than otherwise intended.
I put problem in " " there because this isn't really a problem. It just means that as the GM you may need to adjust some encounters if you want the difficulty at a certain level. Even then, more danger and threat is added from unlucky rolls than from set hit point amounts, so even modifying combat may not be necessary.
Live or Die By The Dice
The old school gamer in me still feels you should stick to the rolls regardless. This is fine too, but keep in mind if someone rolls really low for hit points, they're going to go down a lot faster than you - or the game - will otherwise expect. Just as many adjustments need to be made to keep difficulty on par for below strength PCs as above strength PCs.
Be aware of what this - and the other options - can mean when choosing how to do hit points in your game. Go for the way that fits the feel you want. Just be aware what that fix can mean.