Talk:Fate/EXTRA CCC Collaboration Event/@comment-26948189-20170427013134/@comment-31096423-20170427134620

If you want to get an almost guaranteed roll for the 5 star, the rate being a mere 1%, it would require around 500 attempts to get an almost 98% chance on the next attempt to get a 5 star (reverse statistics calculation for continuus attempts)

How this is calculated, well, here's this example:

1% = 0,01 chances out of 1, if we reverse this, it means we have a 0,99 out of 1 chances of failing, if we multiply this for the next attempt, it will give us the chances of failing to get the desired result in the next attempt. 0,99 chances of failing multiplied by 0,99 chances of failing on the next attempt (since the actual rate does not change) = 0,99 x 0,99 = 0,9801 chances of failing on the second attempt = 1 - 0,9801 = 0,0199 chances out of 1 to succeed on the second attempt = 1,99% draw rate for a 5 star on the second attempt.

If we keep up with this calculation, on the 100th attempt, we get this number: 0,99^100 = 0,36603 = 36,603% of failing = 63,397% chances to draw a 5 star servant on the 100th attempt. In order to get an almost 100% (which is impossible), the number of draws nears 1.000. 0,99^500 = 0,00657 => 1 - 0,00657 = 0,99343 = 99,343% chances of drawing a 5 star on the 500th attempt.

In other words, we can conclude that, in order to get a 5 star servant, we need to at least roll 500 cards, which is around 50 10 grand summon attempts. Do note that the 10 Grand Summon has higher chances of providing a 5 star than single attempts, though it's a very small increase.

Now, we have calculated how to almost certainly get a 5 star servant, but this does not mean we will get the servant of our choice. DW has not disclosed what the "rate up" actually translates to in terms of draw rates, but from my observation, this should almost be near 50% (at least I wishfully think this way to not get depressed). This means that for every 5 star servant we manage to pull, only 1 out of 2 will be the one we want; this being the case, the number of draws increases a bit...

Now the number are as follows: Chance of getting a 5 star from the rate up = 0,5% = 0,005 out of 1 => 1 - 0,005 = 0,995 = 99,5% chances of failing to get the desired draw.

0,995^100 = 0,60577 => 1 - 0,60577 = 0,39423 = 39,423% rate of getting a 5 star from the rate up on the 100th attempt.

0,995^500 = 0,08157 => 1 - 0,08157 = 0,91843 = 91,843% rate of getting a 5 star from the rate up on the 500th attempt.

0,995^1000 = 0,00665 => 1 - 0,00665 = 0,99335 = 99,335% rate of getting a 5 star from the rate up on the 1000th attempt.

And that's with a 50% on the rate up, but I secretly believe the rate is simply double of the normal 5 stars within the pool...

So, there you have it, 500 pulls for an almost guaranteed 5 star, and 1000 pulls for an almost guaranteed 5 star from the rate up. And remember, doing the 10 grand summon means the number of pulls should be lower. Also, do keep in mind that these numbers are only draw rates and are an aproximation, in theory, a player could spend trillions of dollars and NEVER, EVER get a 5 star, and, conversly, it also means that someone could potentially pull 5 stars, always. This means that the numbers shown above are for the unlucky people, the so called E Rank luck players, and that's the number of pulls they should expect to do in the game.