Both hands in the right half of the dial

Question

You glance at an ordinary analogue clock at a uniformly random moment during a 12-hour period. The "right half" of the dial is the half you sweep through going clockwise from the 12 to the 6.

What is the probability that both hands are in the right half at the moment you look?

Then a caution: would the same reasoning give an exact answer to "what is the probability that the minute hand is closer to the 12 than the hour hand is"?

Accepted Answer

Exactly 1/4. The hour hand. It is in the right half of the dial precisely from 12:00 until 6:00, which is exactly half of the 12-hour period. The minute hand. Within any hour it is in the right half during the first 30 minutes, from :00 to :30, and in the left half during the second 30 minutes. That is exactly half of every hour, so exactly half of the 12:00 to 6:00 window as well. Combining. Both conditions hold during the first half of each of the six hours from 12:00 to 6:00: six blocks of 30 minutes, which is 3 hours out of 12. Probability 3/12 = 1/4. No approximation was needed: the hour…

Both hands in the right half of the dial

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