150 or 151? I don’t have a strong intuition. I’m inclined to trust your 150, but my intuition says that maybe 151 is right because 100+99/2+almost1 rounds up to 151. Would have to think about it.
(By the way, I’m not very good at math. (Edit: Ok, fair. Poorly written. What I meant is that I have not obtained certain understandings of mathematical things that those with formal educations in math have widely come to understand, and this leads me to being lower skilled at solving certain math problems than those who have already understood certain math ideas, despite my possibly having equal or even superior natural propensity for understanding math ideas.). I know high school math plus I took differential equations and linear algebra while studying mechanical engineering. But I don’t remember any of it well and don’t do engineering now or use math in my work. (I do like forecasting as a hobby and think about statistics and probability in that context a lot.) I wouldn’t be able to follow your math in your post without a lot of effort, so I didn’t try.)
Re the almost1 and a confusion I noticed when writing my previous comment:
Re my:
E.g. For four 100s: Ctrl+f “100,100,100,100” in your mind. Half the time it will be proceeded by an odd number for length 4, a quarter of the time it will be length 5, etc.
Since 1/2+1/4+1/8...=1, the above would seem to suggest that for four 100s in a row (or two 6s in a row) the expected number of rolls conditional on all even is 5 (or 3). But I saw from your post that it was more like 2.72, not 3, so what is wrong with the suggestion?
I thought of the reason independently: it’s that if the number before 66 is not odd, but even instead, it must be either 2 or 4, since if it was 6 then the sequence would have had a double 6 one digit earlier.
150 or 151? I don’t have a strong intuition. I’m inclined to trust your 150, but my intuition says that maybe 151 is right because 100+99/2+almost1 rounds up to 151. Would have to think about it.
(By the way, I’m not very good at math. (Edit: Ok, fair. Poorly written. What I meant is that I have not obtained certain understandings of mathematical things that those with formal educations in math have widely come to understand, and this leads me to being lower skilled at solving certain math problems than those who have already understood certain math ideas, despite my possibly having equal or even superior natural propensity for understanding math ideas.). I know high school math plus I took differential equations and linear algebra while studying mechanical engineering. But I don’t remember any of it well and don’t do engineering now or use math in my work. (I do like forecasting as a hobby and think about statistics and probability in that context a lot.) I wouldn’t be able to follow your math in your post without a lot of effort, so I didn’t try.)
Re the almost1 and a confusion I noticed when writing my previous comment:
Re my:
Since 1/2+1/4+1/8...=1, the above would seem to suggest that for four 100s in a row (or two 6s in a row) the expected number of rolls conditional on all even is 5 (or 3). But I saw from your post that it was more like 2.72, not 3, so what is wrong with the suggestion?
There is an important nuance that makes it ~n+4/5 for large n (instead of n+1), but I’d have to think a bit to remember what it was and give a nice little explanation. If you can decipher this comment thread, it’s somewhat explained there: https://old.reddit.com/r/mathriddles/comments/17kuong/you_roll_a_die_until_you_get_n_1s_in_a_row/k7edj6l/
I thought of the reason independently: it’s that if the number before 66 is not odd, but even instead, it must be either 2 or 4, since if it was 6 then the sequence would have had a double 6 one digit earlier.