= Solution
Under the no-death assumption, use the mixture of the two community-service paths. Of 10,000 initially assigned offenders, 4,000 breach and leave their subsequent 60-day prison term at day 120. Their remaining 240 days of follow-up match the supplied reconviction window. The 6,000 completers finish community service at day 180 and have the supplied 180-day post-completion follow-up.
The <expected value> of the number reconvicted is consequently
$$
\mathbb E R=4000(0.80)+6000(0.40)=3200+2400,
\qquad
\boxed{\mathbb E R=5600.}
$$
Thus the initial-policy 360-day reconviction <probability> is $0.4(0.8)+0.6(0.4)=0.56$. This is a mixture calculation for the initially sentenced cohort, not the 40% rate of its selected completers alone. It uses the supplied windows and assumes they capture the reconvictions being counted, rather than adding unspecified reconvictions during the earlier order or custody periods.
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