Person A selects a number $X$ randomly from the uniform distribution $[0,1]$. Then Person B repeatedly, and independently draws numbers $Y_1, Y_2, Y_3, Y_4...$ from the uniform distribution on [0,1] until he gets a number larger than $ \frac X2 $ then stops. The expected number of draws that person B makes equal to ?
So I know what uniform distributions are but I have never encountered such a question before. Someone told me to use iterated expectations, but I dont know what they are and how to use them.
Denote by $Y$ the number of draws person $B$ makes. $Y\vert X$ is a Geometric random variable with probability $1-\tfrac{X}{2}$, so $E(Y\vert X)=\tfrac{1}{1-\tfrac{X}{2}}=\tfrac{2}{2-X}$. We want $E(Y)$ so we can use the law of total expectation: $$E(Y)=E(E(Y\vert X))=E(\tfrac{2}{2-X})=\int_0^1\tfrac{2}{2-x}dx=\ldots=\ln(4)$$