Once again, you are working in an adjacent room when your partner walks out and hands you the baby monitor. You completely lose track of where in the day this happens. You continue working for another $30$ minutes, and this time you do not hear the baby cry. What is the probability that the next time your baby cries they will be hungry?
Now your baby allows you to get at least $30$ minutes of work in, so we know that either: (a) the parental handoff was in the first half hour of one of the $4$ playtimes, in which case the baby's next cry will be for sleep; or, (b) the parental handoff was in the first $1.5$ hours of the one of the $4$ naptimes, in which case the baby's next cry will be for food. This gives a $3:1$ ratio of cry for hunger versus cry for sleep, so the probability that the next time your baby cries will be for hunger is $75\%$.
In this mode, if you work $t$ hours (for some $t \lt 1$) without hearing the baby cry, then the handoff could have happened in the first $2-t$ hours of one of the naptimes, or in the $1-t$ hours of one of the playtimes. If you are able to work for $t \in (1,2)$ hours then again you have $100\%$ clarity that your baby's next cry will be from hunger. So the probability that the next time your baby cries for hunger given that you've already worked $t$ hours without crying is $$p(t) = \frac{2-t}{2 - t + \max\{0,1-t\}},$$ for $t \in [0,2].$
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