# Random Squares

Probability Level 2

A factory manufactures wooden squares whose side length should be one meter. However, the side length of the squares is not always $\SI{1}{\meter}$ but varies uniformly between $\SI{1}{\meter}$ and $\SI{1.1}{\meter}.$ Note that the height and the breadth of each square piece remain equal to each other.

Two inspectors, Alice and Bob, want to calculate the average area of a square.

• Alice claims that the average area is $\SI{1.1025}{\meter\squared}$ because the average side length is $\SI{1.05}{\meter}$ and $1.05 \times 1.05 = 1.1025.$
• Bob claims that the average area is $\SI{1.1050}{\meter\squared}$ because the the area is between $(\SI{1}{\meter} \times \SI{1}{\meter})$ and $(\SI{1.1}{\meter} \times \SI{1.1}{\meter}),$ and $\frac{1 \times 1 + 1.1 \times 1.1}{2} = 1.105$.

Who is correct?

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