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Omni-MATH / Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels…

Problem

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problem

Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels 1,2,…,71,2, \ldots, 7, and the label on the eighth side can be changed and begins as 1. She rolls it several times, until each of 1,2,…,71,2, \ldots, 7 appears at least once. After each roll, if kk is the smallest positive integer that she has not rolled so far, she relabels the eighth side with kk. The probability that 7 is the last number she rolls is ab\frac{a}{b}, where aa and bb are relatively prime positive integers. Compute 100a+b100 a+b.
Plain-text mathematical notation (without MathML)
Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels 1,2,…,7, and the label on the eighth side can be changed and begins as 1. She rolls it several times, until each of 1,2,…,7 appears at least once. After each roll, if k is the smallest positive integer that she has not rolled so far, she relabels the eighth side with k. The probability that 7 is the last number she rolls is (a)/(b), where a and b are relatively prime positive integers. Compute 100a+b.
Original LaTeX notation
Teresa the bunny has a fair 8-sided die. Seven of its sides have fixed labels $1,2, \ldots, 7$, and the label on the eighth side can be changed and begins as 1. She rolls it several times, until each of $1,2, \ldots, 7$ appears at least once. After each roll, if $k$ is the smallest positive integer that she has not rolled so far, she relabels the eighth side with $k$. The probability that 7 is the last number she rolls is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$.

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Source and history

task_id
c0555909-62f2-5131-b08d-d2a092c5a5e3
task_key
test--c0555909-62f2-5131-b08d-d2a092c5a5e3
task_revision_id
1
config / split
/ test

Official source

initial import