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Omni-MATH / Let p be a real number between 0 and 1. Jocelin has a coin that lands heads with probability p and tails with probability 1−p; she also has a number written on a blackboard. Each m…

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problem

Let pp be a real number between 0 and 1. Jocelin has a coin that lands heads with probability pp and tails with probability 1−p1-p; she also has a number written on a blackboard. Each minute, she flips the coin, and if it lands heads, she replaces the number xx on the blackboard with 3x+13 x+1; if it lands tails she replaces it with x/2x / 2. Given that there are constants a,ba, b such that the expected value of the value written on the blackboard after tt minutes can be written as at+ba t+b for all positive integers tt, compute pp.
Plain-text mathematical notation (without MathML)
Let p be a real number between 0 and 1. Jocelin has a coin that lands heads with probability p and tails with probability 1−p; she also has a number written on a blackboard. Each minute, she flips the coin, and if it lands heads, she replaces the number x on the blackboard with 3x+1; if it lands tails she replaces it with x/2. Given that there are constants a,b such that the expected value of the value written on the blackboard after t minutes can be written as at+b for all positive integers t, compute p.
Original LaTeX notation
Let $p$ be a real number between 0 and 1. Jocelin has a coin that lands heads with probability $p$ and tails with probability $1-p$; she also has a number written on a blackboard. Each minute, she flips the coin, and if it lands heads, she replaces the number $x$ on the blackboard with $3 x+1$; if it lands tails she replaces it with $x / 2$. Given that there are constants $a, b$ such that the expected value of the value written on the blackboard after $t$ minutes can be written as $a t+b$ for all positive integers $t$, compute $p$.

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Source and history

task_id
0ef07fb0-631c-5d5b-b9de-2ef2c7b7a8de
task_key
test--0ef07fb0-631c-5d5b-b9de-2ef2c7b7a8de
task_revision_id
1
config / split
/ test

Official source

initial import