A fly lands on the face of a clock whose minute hand extends to the edge of the face and whose hour hand is...

probability/fly.on.clock New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A fly lands on the face of a clock whose minute hand extends to the
edge of the face and whose hour hand is half the length of the minute
hand.  What is the probability that the fly has landed in the triangle
formed by the two hands?

-- Joseph Riel <Joseph.Riel@SanDiegoCA.NCR.COM>

Show solution

In article <noelh-2008962111480001@news.hooked.net>, Noel Hancock
<noelh@hooked.net> wrote:
}Length of the hour hand makes no difference.  50/50.

Even supposing that the hour hand was as long as the minute hand, at no
time does that triangle ever cover half the face of the clock.  I
assume there's a shortcut, but I don't see it.  The area at any given
moment is (h * sin t * m)/2, where t = the angle between the hands, h
is the hour hand and m is the minute hand.  Set m to 1, h to 1/2, and
we get (sin t)/4.  If we assume that t is evenly distributed (an
assumption I'm not sure is valid, the average area is 1/(2pi).  The
area of the clock face is pi.  So the probability of the fly landing in
the triangle is 1/(2*pi*pi).  Cheating and running a computer
simulation gives this result also.

Of course, if I'd been cleverer I would have seen that the average area
was h*m * 1/pi, and the probability h * m * 1/(pi*pi), provided h and m
are both less than or equal to 1.

ObPuzzle:
Show that t is evenly distributed.

-- Matthew T. Russotto <russotto@pond.com>

[ Editor: no answer to the ObPuzzle was found in the collected
postings. ]