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When one transaction is broadcasted, what is the probability that it won't be confirmed? (By probability, I mean the observed probability on the blockchain. If a person intentionally spend twice 1 bitcoin, I imagine that the probability is roughly 0.5). What are the reasons that the transaction will be invalid.

  • one of the reason is double spending.

Now since every 10 minutes, a block is mined. After 10 minutes, what is the probability that the transaction will be invalid. I read that 6 confirmations (so 6 blocks) is a good number.

Now after 60 minutes, what is the probability of invalidating the transaction?

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I'm not sure it's exactly what you're asking about, but pages seven and eight of the Bitcoin white paper are concerned with calculating the probability that an attacker could, after having sent payment to a recipient, change the transaction to instead pay himself - specifically, from the paragraph that begins, "We now consider how long the recipient of a new transaction needs to wait before being sufficiently certain the sender can't change the transaction," onwards.

https://bitcoin.org/bitcoin.pdf

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