= Solution
If the outcome is the first successful binding to \i[any] operator, ten well-separated functionally independent unoccupied targets give ten times the total capture rate. Hence
$$
\boxed{\tau_{3,10}\simeq\frac{V}{30\times10\times k_a}=\frac{\tau_3}{10},}
$$
or roughly $6$–$12\,\mathrm{ms}$ for the estimates above. This assumes nonoverlapping capture zones and no significant depletion before the first event. The first-arrival time at one designated operator is not divided by ten merely because other sites are added, and filling all ten sites is a different occupancy problem.
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