Question 75
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 $p$.