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It's Pi Day at the pizzeria, and orders are flying. In the chaos, you randomly toss a pie into the wood-fired oven, missing that an identical pie is already cooking in there. Although the huge oven is built for many pies, the new pizza lands partially on top of the first one. What is the probability that the center of the top pie is directly over the bottom pie instead of over the bare oven surface?
The probability that the center of the top pie is directly above the bottom pie is 1⁄4. Call the radius of the pizzas r. In the figure below, the solid-line circle represents the bottom pie. Extending its radius out by an additional r defines a second circle depicted with a dashed line. If the center of the top pie lands anywhere inside of this dashed circle, then the two pies will overlap. If it lands outside of the dashed circle, then they won’t. To see why, notice that if the center of the top pie were exactly on the edge of the dashed circle, then because it has radius r, the two pies will be tangent and touch at exactly one point.
Amanda Montañez
We are told that the second pie overlaps the first pie. So essentially the problem asks: What is the probability that a randomly chosen point inside the dashed circle is also inside of the solid circle? This equals the area of the solid circle divided by the area of the dashed circle.
πr2/π(2r)2 = 1⁄4
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