Conditional mean of a bounded success time ($\mathbb{E}[K\mid K\le m]=\frac{{\sum }_{j=1}^{m}j{h}_j{\prod }_{i<j}(1-h_i)}{1-{\prod }_{j=1}^m(1-h_j)}$)

Let $h_j$ be the probability of first success on attempt $j$, conditional on reaching that attempt. Then the chain rule for probabilities gives $P(K=j)=h_j{\prod }_{i<j}(1-h_i)$; no assumption of independent attempts is needed when these are the specified conditional probabilities. The chance of success by attempt $m$ is $1-{\prod }_{j=1}^m(1-h_j)$. Divide the truncated first moment by that probability to obtain the displayed conditional expectation, provided its denominator is positive. This is different from counting all unsuccessful cases as though they succeeded on the last permitted attempt.