1. Suppose you throw darts at a square dart board that is one foot on a side. A
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Question
1. Suppose you throw darts at a square dart board that is one foot on a side. A circular target is inscribed within the square dart board (Figure 1). Your darts are guaranteed to land inside the square, but their location within the square is random (every location is equally likely to be hit by a dart.) What is the probability that a randomly thrown dart lands inside the circular target?
a. Event A occurs when a dart lands in the circle and all possible outcomes correspond to a dart landing anywhere in the square. Therefore, in this case the probability of event A occurring is
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Show that the probability that a randomly thrown dart lands inside the circular target is pi/4 = 0.785 (better than 3 out of4). Calculus is not needed.
b. Suppose the circular target in part (a) has a radius of 0.25 ft. and lies entirely within the square dart board (one foot on a side). What is the probability that a randomly thrown dart lands inside the circular target? Does the probability depend on the location of the circular target within the square? Explain.
2. A circular target is inscribed in a dart board that is an equilateral triangle (Figure 2)
a. Note that the sides of the equilateral triangle have length l. Verify that the coordinates in Figure 2 are correct.
b. Find the areas of the triangle and circle (calculus is not needed.)
c. What is the probability that a dart, which is equally likely to land at any point inside the triangle, lands inside the circle?
3. A square dart board in the coordinate plane has vertices at (
Explanation / Answer
1) probability for landing inside circle = pi (a/2)^2 / a^2 = pi / 4
for radius = 0.25 ft
probability = pi * .25^2 / 1^2 = .0625 pi
it doesnt depend on location of circle inside square as it area probability , wherever the circle is .... inside square probability will be same
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