| Let (2008) | |||||||||||||
Abstract | |||||||||||||
| A short proof is not as satisfying as a long one. A good mathematical proof should do much more than just prove the theorem. It should amuse, instruct, and satisfy our deeper love of knowledge, which is much more important than our mere desire to know whether the theorem is true. I find that the standard proof that the number of elements of Wn: = {w = w1... wn | wi = H or T} equals 2 n is not quite satisfactory. It is based on the very superficial ‘structure theorem’: Wn = Wn−1 × {H, T} which immediately implies that |Wn | = |Wn−1 | · 2, which immediately implies |Wn | = 2 n, by induction. In this note I will give a new proof of this famous fact, that is based on a much deeper and more interesting structure that the set of sequences of Heads and Tails possesses. As a bonus, we will also get a new proof of Erik Sparre Andersen’s famous ‘arcsine ’ result ([F]), that the number of ways of tossing a coin 2n times, and having at least as many Heads as Tails at exactly 2k of these tosses, is � 2n − 2k n − k | |||||||||||||
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