= Solution
The time already elapsed since the noise is $32$ minutes. Conditional on a particular noise time $T=t$ and on still being in progress, <memorylessness of the exponential distribution> gives
$$
P(S-t-32>u\mid S>t+32)=e^{-u/\bar s},\qquad u\geq0.
$$
This law does not depend on $t$. Averaging over the <conditional distribution> of $T$ therefore leaves the same remaining-time law. Hence
$$
\boxed{\mathbb E[m]=\bar s\ \text{minutes}.}
$$
The elapsed $32$ minutes are not subtracted from the original <mean>; the exponential residual lifetime has the same <mean> after any elapsed duration.
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