## 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

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

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

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

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.**