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Omni-MATH / Marisa has a collection of 2⁸−1=255 distinct nonempty subsets of {1,2,3,4,5,6,7,8}. For each step she takes two subsets chosen uniformly at random from the collection, and replaces…

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problem

Marisa has a collection of 28−1=2552^{8}-1=255 distinct nonempty subsets of {1,2,3,4,5,6,7,8}\{1,2,3,4,5,6,7,8\}. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process 28−2=2542^{8}-2=254 times until there is only one set left in the collection. What is the expected size of this set?
Plain-text mathematical notation (without MathML)
Marisa has a collection of 2⁸−1=255 distinct nonempty subsets of {1,2,3,4,5,6,7,8}. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process 2⁸−2=254 times until there is only one set left in the collection. What is the expected size of this set?
Original LaTeX notation
Marisa has a collection of $2^{8}-1=255$ distinct nonempty subsets of $\{1,2,3,4,5,6,7,8\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process $2^{8}-2=254$ times until there is only one set left in the collection. What is the expected size of this set?

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Source and history

task_id
37e1fb44-d413-5afa-b7d3-9ceb21bac1a0
task_key
test--37e1fb44-d413-5afa-b7d3-9ceb21bac1a0
task_revision_id
1
config / split
/ test

Official source

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