Friday puzzle -- probability and geometry

Re: Friday puzzle -- probability and geometry

That's about three button presses on a calculator. (or a quick peek at one of those tricky-numetric charts)

Maybe

but I don't understand your answer. Can you give it numerically without giving away the solution?

Re: Friday puzzle -- probability and geometry

Define random ;-) I can see at least 3 different, equally valid answers depending on the definition of a random chord.

Re: Maybe

The sum of the first 4 digits past the decimal point would be 10 less than the angle that it is the cosine of.

Re: Friday puzzle -- probability and geometry

I think you can get to the answer by considering an inscribed hexagon.

:-)

color me dense

I can't figure out your numeric answer. It may be AN answer, although Dan got THE answer.

Zero if I read it correctly?

I had to travel in the wayback machine to fact check this, but the longest Chord in a circle ithe diameter, so you could never have one that was LONGER...

Am I reading this right?

S

Re: color me dense

I assumed (I know, Never assume.) that a 'random chord' would have an equal probability of passing anywhere between 0 and 1r from the center of the circle. for the chord to be longer than r, the angle between the midpoint of the chord and either endpoint must be >30 degrees (sin(30) = .5).

So, any chord passing between 0 and cos(30) (0.8660) of the center would have a length longer than r.

question was re radius, not diameter

Re: Friday puzzle -- probability and geometry

B flat

OK,

Any time that the chord endpoints form a triangle with the circle's midpoint where the central angle is >60 degrees. At 60 degrees, we are equilateral, and we all know that triangles only have 180 degrees...

s

Solution -- probability and geometry

Dan nailed this one. The problem is not well-defined. When we talk about a "random object", we are really talking about an object chosen by a random process. Usually, one process is so obvious that we let the distinction blur. This problem illustrates why that can be important.

Dan mentioned at least 3 possibilities. I'll show four here, with resulting answers ranging from 1/2 to 86.6%. If miss one of his, maybe he will elaborate.

If you react to this the way I did, as you wrap your mind around this, random processes other than the one you envisioned will seem to introduce a bias away from your "true" answer. But challenge yourself to give an OBJECTIVE argument for your process.

A chord can be defined in a number of ways, leading to obvious random distributions. Each of these processes define all possible chords.

A) Choose two points on the circumference of the circle (evenly distributed) as the endpoints of the chord. If those points are more than 60 degrees apart, the condition is met. Answer = 2/3. This was the "natural" answer for me. Another process leading to the same answer: A') pick one point at random along the circumference let its angle be evenly distributed.

B) Throw a dart to pick a point in the interior of the circle, and draw the chord of which that point is the midpoint. The chord will be longer than the radius iff the point is closer to the center than sqrt(3)/2 times the radius, i.e., if it is inside the circle of radius sqrt(3)/2 as big as the primary cicle. This smaller circle has an area 3/4 as large as the big circle, so there is a 75% chance of the dart hitting in that area and picking (at random) a chord longer than the radius of the big circle. This one "feels" right to me as well.

C) define the chord by its orientation (0-360 degrees, evenly distributed, although that doesn't matter) and its distance from the center of the circle. As in B, it will be larger than the radius if it is closer to the center than sqrt(3)/2. But here we are choosing that distance directly rather than by hitting an area, so the probability is sqrt(3)/2 = 86.6%. This one "feels" less natural to me, although it was the send idea I hit on in trying to find the "right" answer. It is Bill's answer. I can't see any sound reason for rejecting it.

D) This one "begs the question", but is there anything wrong with it? Chose an orientation, and a length, from 0-2r. Answer = 50%.

P.S., I recognize that in C and D my processes got a little sloppy and defined pairs of chords. I left the coin-flip part of those processes out for simplicity ;-)