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# Probability problem

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The dartboard in the diagram is composed of two concentric circles. The radius of the larger circle is twice as long as that of the smaller circle. The dartboard is further divided into four equal parts by two diameters of the larger circle. 200 darts are randomly thrown towards the dart board and 80% of them land off the dartboard.

What is the expected number of darts that land in the blue region?

Dec 9, 2021

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Since 80% of the darts land outside the dartboard only 20% of them land on the dartboard:  20% of 200 = 40 darts.

One-fourth of these darts are expected to land in the upper-right quadrant:  ¼ x 40  =  10 darts.

If the radius of the inner circle is 1, the area of the inner circle is  pi · 12  =  1 pi.

Then the radius of the dartboard is 2 and the area of the full dartboard is  pi · 22  =  4 pi.

The area of the outer circle is  4 pi - 1 pi  =  3 pi.

The ratio of the area of the outer circle to the area of the inner circle is  3 pi / 1 pi  =  3 / 1.

Therefore three-fourths of the darts that land in the upper-right quadrant will land in the shaded area

=  3/4 x 10  =  7.5 darts.

Dec 9, 2021