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MMLU-Pro / 8409 / Suppose that X₁,X₂,... are real numbers between 0 and 1 that are chosen…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Suppose that X1,X2,...X_1,X_2,... are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let S=∑i=1kXi/2iS=\sum_{i=1}^k X_i/2^i, where kk is the least positive integer such that Xk<Xk+1X_k<X_{k+1}, or k=∞k=\infty if there is no such integer. Find the expected value of S.
Plain-text mathematical notation (without MathML)
Suppose that X₁,X₂,... are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let S=∑_(i=1)^(k)X_(i)/2^(i), where k is the least positive integer such that X_(k)<X_(k+1), or k=∞ if there is no such integer. Find the expected value of S.
Original LaTeX notation
Suppose that $X_1,X_2,...$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S=\sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S.

category

math

options

  1. 0.35462541
  2. 0.32876543
  3. 0.25012345
  4. 0.42365489
  5. 0.50567891
  6. 0.28934765
  7. 0.37654298
  8. 0.46789234
  9. 0.29744254
  10. 0.41123567

src

theoremQA-Math

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Source and history

upstream_id / Original problem ID
8409
task_id
bd0997e3-8517-5cd4-80ba-8bb6f3de4a8f
task_key
default--test--8409
task_revision_id
1
config / split
default / test

Official source

initial import