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What are the odds that three points on a circle fall in a semicircle?

Answer: three to one (3:1) or three out of four (3/4), which are two ways of saying the same thing. (Three hits to one miss means a hit happens three times out of four.)

Note: it doesn't matter if the points are on the circumference or in the interior of the circle. If you divide a pie in thirds, then choose one of those thirds, the odds that a point on the circumference will fall in that third are the same as the odds that a point anywhere on the pie will fall in that third. The probability is one out of three, or one-third, or one to two (one hit for two misses).

Note: a similar but different question is this -- What are the odds that if we divide a circle into two halves, say left and right, that three points will land in the left half? The answer to this is 1/2 x 1/2 x 1/2 which is 1/8. It is the same as asking the odds a coin flip will give tails three times in a row......... The question we are asking is different and trickier. It is what are the odds that three points on a circle will land such that they will fall within any possible semicircle.

An intuitive solution.

Two points will always fit in a semicircle. If the two points are exactly opposite they still qualify. So then the issue is how often will a third points fit with the other two in a semicircle. If the two points are close (lets put the points on the circumference for simplicity), then the third point is almost always good. If the two points are nearly opposite, nearly half of the circle is good, the rest not so.  These two extremes for the two points, far apart and close together are equally probable extremes, from 50% to 100% with an even rise in the probability as the two points go from far apart to close together. So the average probability is half way between 1/2 and 1, or 3/4.