Calculating the Odds for Toast’s Yogg OTK

hearthstone 2 - Calculating the Odds for Toast's Yogg OTK

I don't know if anyone's done this yet, but I would just like to do it here for fun.

So DisguisedToast, in one of his latest streams, pulled off a wild combo where he went Shadow Essence into Mechathun (created by Prince Malchezzar) into myra into cataclysm. He claimed that it was a "one in a billion chance" – is this true? I will attempt to calculate the odds to see if the title was true or just clickbait.

The hearthstone wiki tells me that there are 508 spells in wild. He cast 20 spells that game, so the probability of getting Shadow essence, Myra, and Cataclysm in any order would already be 1/(508C3) * 20C3 = 1/19053. Getting Shadow essence, myra, and cataclysm in that particular order would be the previous answer divided by 3!, or 1/114318. Furthermore, there are 237 legendary minions in wild, so the probability of the opponent getting a mechathun from malchezzar will be 1 – (236/237)^5 = 0.0209197619. Since the opponent was quest priest, it would be safe to assume that his starting deck contained at least 13 minions (7 for quest + 5 malchezzar + malchezzar himself), so using this as a rough estimate for minions remaining in the opponent's deck, we get 1/13 chance for shadow essence to pick out mechathun.

So the total probabability would be 1/114318 * 0.0209197619 * 1/13 = 0.0000000140766323 ≈ 14/1,000,000,000, so it seems as though the title was just clickbait, at least for my standards.


However, there are some spells that Yogg can cast to cancel itself, and there are 90 spells that can, on their own, cancel the battlecry. There are some other combinations of spells that also cancel, but there are also other combinations that nullify this so I did not include this in my calculations. So the probability that the Yogg will survive to complete its battlecry and trigger the cataclysm is somewhere around (1-(90/508))^20 = 0.02024190535, or around 1/50. We know that the probability found will actually be slightly less than actual probability of Yogg surviving, but it is not different enough to change the outcome. 14/1000000000 * 1/50 < one in a billion, so the calculations suggest that the probability is, in fact, one in a billion as the video promised.

These calculations are very rough, but even with these rough calculations I hope it's easy to see how much of a RNG lord Toast is.

TL;DR: Toast pulled off an insanely improbably combo:

and I calculated approximately the probability.

Source: Original link

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