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I suppose that "at random from within the circle" means "equal chances for areas with equal areas". Then the probability in question is the ratio of the shaded region to the area of the entire circle.
Entire circle has central angle of 360°. The shaded region has the same area as a sector with the central angle of 109°+83°=192°. The area of a sector is directly proportional to its central angle.
Therefore the ratio of areas and the probability is
`192/360 = 8/15 approx 0.5333.`
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