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A philosopher of science examines falsifiability. A theory claims All swans are white. After observing 10,000 white swans but one black swan, by what factor does the falsity probability increase if the model assumes universal statements with no exceptions? (Assume initial falsity probability is 0.01, and black swan doubles it.)
Alternative interpretation: If one counterexample doubles the chance of falsification.
Solution: Substitute $ x = 2 $ into both functions:
Given $ f(2) = g(2) - 10 $, substitute:
Solution: Substitute $ x = 5 $ into $ f(x) $:
Question: Find the $ y $-intercept of the line passing through $ (1, 4) $ and $ (3, 10) $.
Solution: Calculate the slope $ m $:
Use point-slope form with $ (1, 4) $:
Question: A historical artifact features a square with a diagonal of length $ d $. A circle is inscribed in the square. What is the ratio of the circles circumference to the squares perimeter?
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