Audio note: this article contains 247 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

Which of the following do you think is bigger?

<span>_A_</span>: The expected number of rolls of a fair die until you roll two <span>_6text{s}_</span> in a row, given that all rolls were even.

<span>_B_</span>: The expected number of rolls of a fair die until you roll the second <span>_6_</span> (not necessarily in a row), given that all rolls were even.

If you are unfamiliar with conditional expectation, think of it this way: Imagine you were to perform a million sequences of die rolls, stopping each sequence when you roll two <span>_6text{s}_</span> in a row. Then you throw out all the sequences that contain an odd roll. The average number of rolls in the remaining sequences [...]

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Outline:

(01:49) A quick verification

(02:52) Primer on geometric distributions

(05:52) Rolls until first 6, given all even

(10:05) Adding conditions after the first 6

(11:56) Reviewing the paradox

(13:22) Proving _BA_

(17:44) The general case

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First published:
November 22nd, 2024

Source:
https://www.lesswrong.com/posts/7TYdQ34KxTBmA734v/a-very-strange-probability-paradox

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Narrated by TYPE III AUDIO.

“A very strange probability paradox” by notfnofn

LessWrong (30+ Karma)