1. THAT UBIQUITOUS  “irrational number” - cannot be expressed as a fraction m/n of integers.

4. via symmetry of N(0, 1)

5. Comte de Buffon “Needle Problem” Suppose a needle is repeatedly dropped onto a series of equally-spaced parallel lines. Georges-Louis Leclerc “Comte de Buffon” 1707-1788 With what probability will the needle cross one of the lines?

6. Two independent random variables define the needle’s position and orientation: height ( y) of the needle’s midpoint to the closest line, height ( y) of the needle’s midpoint to the closest line,

7. Two independent random variables define the needle’s position and orientation: height ( y) of the needle’s midpoint to the closest line, height ( y) of the needle’s midpoint to the closest line, the acute angle (θ) of the midpoint to the horizontal, the acute angle (θ) of the midpoint to the horizontal, Can reasonably assume that y and θ have uniform distributions over their respective domains.

8. uniform distributions Can reasonably assume that y and θ have uniform distributions over their respective domains.

9. 0 0 uniform distributions Therefore, via independence, independence, the joint probability distribution function function is

10. 0 In order to cross the line, Hence,

11. 0 In order to cross the line, Hence, In particular, if