An unexpected result involving sums of random numbers

[Albert Chan]

If random values are discrete, say coin flips, (head=1, tail=0), the expected value (sum ≥ 1) = 1/p = 2

Code:

        n * P(n)
1       1 * 1/2
01      2 * 1/4
001     3 * 1/8
0001    4 * 1/16
...

This gives a neat prove for E = Σ(n/2^n, n=1 .. Inf) = 1/p = 2
see https://math.stackexchange.com/questions...onverge-to

At the end of each simulation, sum is exactly 1, without "waste".

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https://www.fourmilab.ch/documents/random_sum/ examples ("waste" in bold)

0.4770 + 0.0516 + 0.1793 + 0.2250 + 0.0438 + 0.3006 = 1.2773
0.8474 + 0.3005 = 1.1479
0.5535 + 0.2721 + 0.5390 = 1.3646
...

For continuous uniform distribution of [0, 1), we do have wastage (almost always)

Last random number, on average = 0.5
Balance (before adding last random number), on average, between 0.5 to 1.0

1.0 < E*0.5 < 1.5      → 2.0 < E < 3.0

FYI, above link has the prove for E = e ≈ 2.71828