You win by getting $10$ heads in a row (fair coin). Any other result and you lose immediately. For example if you start flipping and get $T$ or $HT$ or $HHHT$ you lose. Let $X$ be the number of coin flips in this game.
It seems correct to me that $E[X \mid win] = 10$
I know that
$$E[X \mid win] = \frac{E[X \mathbf{1}_{win}]}{P(win)}$$
I'm pretty sure $P(win) = \frac{1}{2^{10}}$ and therefore,
$$E[X \mathbf{1}_{win}] = 10 \cdot \frac{1}{2^{10}} = .00976$$
My specific question: Assuming my numbers above are correct... how would you explain $E[X \mathbf{1}_{win}] = .00976$ in words?
My attempt: I suppose it's $10$ times the probability of winning the game but I'm not sure what to think of that... Functionally speaking I believe $E[X \mathbf{1}_{E}]$ is the regular expected value formula but instead of across all possible values of $X$, you do it for only the values of $X$ in the event $E$. For this coin game I'm having trouble thinking about the result $.00976$.
Motivation: I'm trying to build my understanding of conditional expectation. Thanks for your help!
From your comments and question I get the idea that you want an intuitive explanation of the conditional probability and the expectation with an indicator function in there. I would interpret $\mathbb{E}[X ∣ win]$ as the expected amount I would win when I am sure that I have won. In this case we know that we only win if $X=10$ so this is less interesting, but if we had another function for the payoff of the bet this would also be fine. I'm sure you know what would happen if we were to win as well if $X=$. You surely would be able to compute $\mathbb{E}[X ∣ win]$.
On your second question in the comments: "Has anyone ever run into a word problem or situation where you wanted to know $\mathbb{E}[X1_E]$ as the final answer?" This is a simple generalization of the above, the indicator function does the conditioning on the event $E$.