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 [...]
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
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).
