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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Mr. Fox has dug a tunnel in the hope of breaking into a chicken coop. He is now at a point which is roughly at the same distance from the farms of Bean, Boggis and Bunce. He can see Bean's farm under an angle of , Boggis' farm under an angle of and Bunce's farm under an angle of . Assuming he digs in a random direction, what are the chances of ending up on none of these farms?

(characters from Roald Dahl's "Fantastic Mr. Fox")

Since he is equally far from three different points, he is at the center of a circle. The three farms can be approximated to arcs of this circle. The probability of reaching a farm is directly proportional to the angle that intercepts the arc corresponding to the farm.

figure

The total angle that Mr. Fox can dig under is equal to .

The total angle under which he can reach one of the three farms is:

equation

The probability of reaching one of the farms is:

equation

And the probability of not reaching any of the farms is:

equation