How Air Stocks Are Making Sports Gear Look & Feel Super Fresh—See Why Everyones Obsessed!

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How Air Stocks Are Making Sports Gear Look & Feel Super Fresh—See Why Everyones Obsessed!

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Solution: Let the arrival times of Dr. Tide and Dr. Coral be $ x $ and $ y $, respectively, where $ x, y \in [0, 60] $ minutes after 7:00. The condition $ y > x $ defines a triangular region in the $ xy $-plane with area $ \frac{60^2}{2} = 1800 $. The favorable region is where $ x < 45 $ and $ y > x $. Integrating over $ x \in [0, 45] $, the area is $ \int_{0}^{45} (60 - x) \, dx = 60 \cdot 45 - \frac{45^2}{2} = 2700 - 1012.5 = 1687.5 $. The conditional probability is $ \frac{1687.5}{1800} = \frac{3}{4} $.
\boxed{\dfrac{3}{4}}
Question: A historian studies three scientific breakthroughs from a collection of 20, where 5 are pivotal. What is the probability that exactly two of the selected breakthroughs are pivotal?

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