What Does The Crying Baby Want?

This week’s Riddler Express asks:

As any caretaker can tell you, one challenging part of caring for an infant is interpreting their cry. Are they hungry? Are they tired? Do they need a diaper change?

Suppose you have an infant who 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 several times over the course of a 12-hour day. (Your rock star baby sleeps peacefully 12 hours through the night.)

You’re working in an adjacent room when your partner walks out and hands you the baby monitor. You’ve completely lost track where in the day this happens. You continue working for another 30 minutes, then you hear the baby cry. What’s the probability that your baby is hungry?

Solution

Another fun, easy, but counter-intuitive question. One might think that since the baby spends twice as much time sleeping as playing, they are twice as likely to be hungry as tired. But, to answer this question it can be helpful to map out the schedule visually:

With this diagram we can see that there are eight possible times that correspond with the time your spouse hands you the baby monitor, and four of those are during the nap phase and four during the play phase. When the baby starts crying, there’s a 50% chance it’s because they need to be fed.