Specifically I’m asking about this situation that came up during a D&D game:
Say I’m rolling a 20 sided die, and I do not want to roll a 1. I know that the odds of rolling a 1 are 1/20.
I know that the chances of rolling a 1 twice in a row is (1/20 * 1/20), which is far a lower occurrence.
Say then, before I rolled my “real” roll, I rolled the die again and again until I landed on a 1, then proceeded to roll my “real” roll, would I have reduced the odds of rolling a 1 to (1/20 * 1/20), given that I’ve just rolled a 1 prior?
This is the logic I’m having trouble reasoning about and I’d appreciate it if anyone could clarify what is or is not accurate about the assumptions being made in this scenario.
In: Mathematics
Your die does not “remember” what you rolled last time. Every roll has a 1/20 chance to roll any given number.
If you roll the die X times, however long it takes until you roll a 1, and then roll once more, the chance of a 1 on the “once more” roll is still 1/20…just like it is for every single individual roll you make, ever.
The only reason the odds of rolling back-to-back 1s are lower is because that calculates the chances of **both** rolls being 1s – the first roll has a 1/20 chance for a 1, the second roll has a 1/20 chance for a 1, but the odds that they’ll **both** be 1s is very small. That doesn’t change the fact that each one is still a 1/20 chance **on its own**.
You’re correct that the odds of you rolling 2 1’s in a row are (1/20) * (1/20) but your setup means that it doesn’t help you. Your odds of any given roll landing on a 1 are 1/20.
>before I rolled my “real” roll, I rolled the die again and again until I landed on a 1
This means that you’re not wondering the odds of rolling two 1’s in a row, you’re only looking at the odds of rolling a 1 *after you know you have already rolled 1*, the odds of the getting two 1’s in a row in this case aren’t (1/20) * (1/20) they’re (1) * (1/20) since you’re only calculating odds after you’ve already rolled a 1, its a given so its probability is 100% (you can’t get here if it didn’t already happen)
If you really want to change your odds of rolling a 1, get an automated roller and check the bias of your dice. Most plastic D20 dice are accidentally weighted to one side or another and that changes the real roll distribution
Each individual roll stands on its own. So each time you pick up a 20 sided die, you have a 1 in 20 chance of rolling any of the numbers.
Before you started any roll, the chance of rolling one of the numbers two times in a row is pretty small. But even if you rolled twelve ones in a row, the next time you pick up your die, you still have a 1 in 20 chance in that moment. The past doesn’t alter the odds on your specific roll.
It’s just the odds of rolling 13 ones in a row is incredibly small when you haven’t rolled once yet. But after you’ve rolled 12 ones in a row? Your odds are 1 in 20.
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