Take a Number N between 0 and 1 not inclusive. Then start selecting numbers randomly between 0 and 1 non inclusive. What is the expected amount of numbers I need before My product is less than or equal to N? Using the definition of expected value and some integration I have a formula based on integration and extrapolation of patters but I’m not sure if it’s correct or simplified here is my formula
2026-04-07 21:15:40.1775596540
expected value before product is less than N
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This can be solved as as Markov chain with states $s_0$ and $s_1$. Once you sample a number in the interval $(0,p)$, you are done. Hence: $$ m_{0,1} = 1 + (1-p)m_{0,1}\\ m_{0,1} = \frac{1}{p} $$ where $m_{i,j}$ is the mean first hitting time of state $1$ if you start in state $0$.