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?

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

_B_: The expected number of rolls of a fair die until you roll the second _6_ (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 _6text{s}_ 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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Source:

Narrated by TYPE III AUDIO.