The speedrunner Dream has recently been accused of cheating in his Minecraft Java v1.16 glitchless speedrun.
How did the Minecraft Java speedrunning mods calculate if Dream cheated?
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The Short Summarized Answer
These were drop rates for two items important to the speedrun attempts over the course of six livestreams.
Ender pearls are obtained 4.7% of the time. Out of 262 rolls they were obtained 42 times.
Blaze rods are obtained 50% of the time. Out of 305 rolls the were obtained 211 times.
The probability of being at least this lucky in each case can be modelled by a binomial distribution.
For the ender pearls this probability is 0.00000000000565319.
For the blaze rods this probability is 0.00000000000879143.
These probabilities are independent so the chances of both occurring is their product, 0.000000000000000000000049699624 or 1 in 20 sextillion.
You can do some adjustments for biases but they don't even come close to reaching a reasonable probability.
The Long In-Depth Answer
This video sums up the decision behind the mod's actions, based on a paper made studying how incredibly statistically unlikely his run ended up being. You can either watch the video or read the paper to understand the reasoning behind their actions.
Did Dream cheat?
Yes, he's actually admitted to doing so. Here's the relevant clicky. In a recently deleted tweet he posted a confession of sorts that revealed he had "inadvertently" used mods to enhance the drop rate of ender pearls and blaze rods.
You can find an archived version of his confession here. The relevant paragraph in regards to cheating in his run:
After considering this, I ended up finding out that I HAD actually been using a disallowed modification during ~6 of my live streams on Twitch. At the time we were just starting to record videos on 1.16 and we had just hired a developer to help with coding mods for videos because me and George had no experience with mods only plugins. One of the mods that they were working on was an overall recording mod, that I have used in every video (with updates and improvements) since around the speedrun controversy.