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If I summon 100 times on Halloween banner, will I get Halloween Elisanne?

Probability guide for summoning Elly in 100 pulls.
1 month ago
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If I summon 100 times on Halloween banner, will I get Halloween Elisanne?

Overview

The standard Diamantium packs in the US are $80 for 4,200 Diamantium pack. $320 buys you 100 summons with 1,800 Diamantium left over.

As our math will show through, even burning that much cash only gives coin-flip odds for summoning Halloween Elisanne.

The game doesn’t completely leave everything to luck, however, as there are pity mechanics that prevent losing streaks:

  • 10 Pull Pity Mechanism: Every 10 pull without 5, your overall 5 rate increase by 0.5%.
  • 100 Pull Pity Mechanism: If you fail to get any 5 in 100 pulls, your next summon is a guaranteed 5.

There’s also a guaranteed 4 for 10x pulls, but this does not increase 5 rate.

Worst Case

Worst Case

If the pity rate doesn’t come into play, what are the odds of getting Halloween Elisanne?

In probability, the odds of the desired outcome P over multiple attempts N is calculated as

$$100\% - (100\% - P)^N$$

Instead of P * N to account for scenarios when you get multiple Halloween Elisannes in a set or rolls.

With Halloween Elisanne rate starting at P = 0.5%

$$100\% - (100\% - 0.5\%)^{100} = 39.4\%$$

This can be considered the worst case scenario, where the 10 Pull Pity Mechanism never kicks in because you’re constantly pulling 5★ Wyrmprints instead.

Best Case

Best Case

The best case scenario, where your first 5★ pull is Halloween Elisanne.

In this scenario, the odds of getting Halloween Elisanne increases 0.0625% every 10 pulls, which increases your overall odds by:

10 Pull # Get Elly Not Get Elly
1 4.89% 95.11%
2 5.22% 94.52%
3 5.74% 93.92%
4 6.26% 93.33%
5 6.77% 92.75%
6 7.27% 92.17%
7 7.75% 91.59%
8 8.23% 91.01%
9 8.70% 90.44%
10 9.16% 89.87%

In the best case scenario, your chance of summoning Halloween Elisanne is 70%.

Because your Pity Rate resets when you summon any other 5, your actual odds will be somewhere between the two scenarios: 39.4% ~ 70%.

Typical Case

Typical Case

More likely, your 5 rate resets every two 10 pulls, as such your Elisanne rate oscillates between 0.5% and 0.5625% instead.

10 Pull # Get Elly Not Get Elly
1 4.89% 95.11%
2 5.22% 94.52%
3 4.62% 95.11%
4 5.22% 94.52%
5 4.62% 95.11%
6 5.22% 94.52%
7 4.62% 95.11%
8 5.22% 94.52%
9 4.62% 95.11%
10 5.22% 94.52%

In the typical scenario, your chance of summoning Halloween Elisanne is about 50%.

Best Worst Case

Best Worst Case

Let's assume you're really unlucky: What’s the chance of hitting Halloween Elisanne as the guaranteed 5★ in the 100 Pull Pity Mechanism?

This requires two things to happen:

  • Not hit a single 5★ in 100 rolls
  • Has Halloween Ellisane

The chance of NOT summoning a single 5★ in your 1st 10x would be:

$$(100\% - 4\%)^{10} = 66.5\%$$

5★ rate increase by 0.5% each time, so then 2nd 10x would be:

$$(100\% - 4.5\%)^{10} = 63.1\%$$

And so on:

10 Pull # 5★ Rate Not Get 5★
1 4.0% 66.5%
2 4.5% 63.1%
3 5.0% 59.9%
4 5.5% 56.8%
5 6.0% 53.9%
6 6.5% 51.1%
7 7.0% 48.4%
8 7.5% 45.9%
9 8.0% 43.4%
10 8.5% 41.1%

The odds of never summoning a 5★ in 100 pull then would be:

$$P = 66.5\% \times 63.1\% \times 59.9\% \times 56.8\% \times 53.9\% $$

$$ \times 51.1\% \times 48.4\% \times 45.9\% \times 43.4\% \times 41.1\% = 0.156\%$$

The chance of summoning Halloween Elisanne is 12.5% for the guaranteed 5★ summon (chance of any particular 5★ are proportional).

The odds of these two things happening at the same time would be:

$$P = 0.156\% \times 12.5\% = 0.02\%$$

Your odds of getting Halloween Elisanne via the 100 Pull Pity Mechanism is 1 in 5,000.