Acute Angles and Wall Clocks

 Follow @ProbabilityPuz Q: If you were look at a wall clock between 12noon and 1pm at random, what is the probability that the minute hand and the hour hand make an acute angle?
Kindle Fire HDX 7", HDX Display, Wi-Fi, 16 GB - Includes Special Offers A: If you rush this one, you will be tempted to answer 50%, but it is not so. Assume we are at 12 noon. The angle swept by the hour hand of a clock in an hour is $$\frac{30^{o}}{60}=(\frac{1}{2})^{o}$$ per minute. If $$t$$ minutes have passed in an hour, the hour hand would sweep out $$\frac{1}{2}t$$ degrees. Likewise the minute hand of the clock sweeps out $$\frac{360^{o}}{60} = 6^{o}$$ per minute. The condition required for an acute angle to form between the hour and minute hand can be phrased as
$$\theta_{m} - \theta_{h} \le 90^{o} \\ 6t - \frac{t}{2} \le 90 \\$$
The above simplifies to
$$t \le \frac{180}{11} \approx 16$$
However, there is one more case that creates an acute angle between the hour and minute hand of the clock. This is when the minute hand makes it ways back to 12. This condition can be stated as
$$270^{o} \le \theta_{m} - \theta_{h} \le 360^{o}\\ 270^{o} \le 6t - \frac{t}{2} \le 360^{o}$$
which in turn implies $$t$$ has a range of $$\frac{2}{11}\times (360 - 270) = \frac{180}{11}$$. The above two inequalities shows that there is a total of $$\frac{2\times 180}{11} = 32.72 \text{ minutes}$$ when the hour hand and the minute hand forms an acute angle. Given that there are 60 minutes in an hour the sought probability is $$\frac{32.72}{60} = 54.53\%$$. Notice, at first blush one is tempted to answer 50% as the sought probability, but when one thinks more about it, it is intuitive to see why it will always be slightly greater than 50%. This is because of the hour hand moves slowly too, maintaining the "acuteness", a bit longer. This gives the higher probability number.

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An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition
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Fifty Challenging Probl…

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