Why Is 6 the Number of Confirmations that Is Considered Secure?

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This is proved by a mathematical calculation. As the number of blocks increases the probability of the attacker, with his alternate chain, taking over the honest chain drops exponentially. After 6 confirmations, the race between the honest chain and an attacker chain are substantially big enough that the success event of the attacker’s chain overtaking the honest chain is vanishingly small.

Here are some of the relevant sections from Satoshi's paper: http://bitcoin.org/bitcoin.pdf

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