Need someone to explain the math in getting averages (1% x100 + duplicates)

[Deleted]

Let's say you have 100 rolls and X pops up 1% of the time.

On average you see X at least once is 63.4% (on 200 rolls its like 84%)

but when you factor in the possibility of seeing X twice/thrice/up to 100 times in a row, should you still consider yourself unlucky if you don't see it once in 100 rolls?

[luc54]

The average already consideres cases in which X happen more than once. 63% chance isn't that likely that I'd consider myself very unlucky

[Deleted]

The average already consideres cases in which X happen more than once. 63% chance isn't that likely that I'd consider myself very unlucky

but that 63.4% comes from 1-(chance of it happening at least once). doesnt say anything about duplicates

[luc54]

but that 63.4% comes from 1-(chance of it happening at least once). doesnt say anything about duplicates

It does. P(x happened once) +p(x happened twice)...+p (x happened 100 times) is really hard to calculate so you take a compliment which is 1-p (x never happens)=1-p (X)^100~63%

[Danner]

Let's say you have 100 rolls and X pops up 1% of the time.

On average you see X at least once is 63.4% (on 200 rolls its like 84%)

but when you factor in the possibility of seeing X twice/thrice/up to 100 times in a row, should you still consider yourself unlucky if you don't see it once in 100 rolls?

TLDR Dramatization:
So the Italian Mafia has you up at gunpoint. You're told to roll this 100 sided dice 100 times. All the faces, except one is blank. The last side has a skull on it. If you roll the skull, they will give you a solid pair of cement shoes and a nice moonlight swimming opportunity. The chance of you ending up sleeping with the fishes is 36.6%.

It's not terrific odds. But it's not horrible either.
On average, you should win this gamble. I'd consider myself slightly unlucky if I lost it. But I wouldn't take those odds voluntarily either .

To answer your question:
The 63.4% number in your question already include the chance of rolling 1, 2, 3 or more skulls. It doesn't really matter how many you roll, as rolling a single one is bad for you. Basically, the 63.4% chance is the answer to the question "what are the chances or rolling at least one skull in 100 tries".

The opposite of that is not rolling a skull at all. This is simply given as 100% - 63.4% = 36.6%. This is the answer to "what are the chances of not rolling a skull in 100 tries".
You can also find that through binomic probability. What are the odds of missing the skull on the dice 100 times in a row? Aka 0.99^100 = 36.6%

The Simulation:

I like Anydice. It's great for finding these kind of statistics.
http://anydice.com/program/c700

I created a dice with 100 faces, where 99 are 0 (blank) and one is 1 (skull). I call it the "one percent die". I used these values, because you can now use the value to get the average number of skulls you roll up by simply adding up the die results. And anydice does that well.

If you sum up 100x "one percent die" results, you will on average get the value of 1. Ie, one skull. Simple enough.

If you click on the "At Least" tab, you can see the chance of rolling at least 1 as a sum of 100 of these dice. It is 63.4%, just as you say.
If you want to calculate this manually, it's chance of the sum rolling 1 + the the chance of the sum rolling 2 + the chance of the sum rolling 3, and so on - up to 100. Ie, the sum of any other sum except the outcome where all dice are 0. You can see the chance of rolling a sum of exactly one, two, three and so on under the "Normal" tab.

Under the "Normal" tab, chance of sum=0 is indeed 36.6%. As it should be