baby please don’t cry

First, an express riddle from the Riddler of last week:

An infant naps peacefully for two hours at a time and then wakes up, crying, due to hunger. After eating quickly, the infant plays alone for another hour, and then cries due to tiredness. This cycle repeats over the course of a 12-hour day. (The baby sleeps peacefully 12 hours through the night.) At a random time during the day, you spend 30 minutes with your baby and then the baby cries. What’s the probability that your baby is hungry?

The probabilistic setting is somewhat unclear, in particular because the last daytime nap is followed immediately with a 12 hour night sleep. Or the 12 hour night sleep is immediately followed by a one or two hour nap. Assuming a random starting time over the 12 hour period, denoting X as the time to the next crisis and Y as the nature of the cries (H versus T), it is straightforward to show that P(Y=H|X=30′) is ½. While it would be 1 for any duration larger than one hour.

Followed by an extra one this week:

Starting at a random time, 30 minutes go by with no cries. What is the probability that the next time your baby cries she will be hungry?

Which means computing P(Y=H|X>30′). Equal to ¾ in this case.

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