You are assuming the trials are dependent. That the failure of one hash reduces the search space and increases the chance of future hashes being correct (like removal of cards from blackjack deck). The reality is hashes are independent*. If difficulty is one million the odds of each hash valid are 1/(one million * 2^32). This is more like rolling a pair of fair dice. If you fail to roll "12" ten trillion times in a row the chance of the next roll being "12" is still the same. Likewise if you hash one trillion hashes above the target the odds of the next hash being valid is still 1/(one million * 2^32). Note the expected outcome is just that expected. It is entirely possible that the "long run" will never arrive in your lifetime (or the lifetime of the human race). The actual outcome could be more or less than expected for a nearly infinite amount of time.
* Technically they aren't independent because the search space is finite. However for a given block height (thus prior block hash, difficulty, and version are fixed) the search space is 2^320 which for all practical purposes is infinite. While Bob may complete twice as many hashes before the block both Bob and Alice will complete roughly ~0% of the search space.
Topic: Does 2x hashing rate mean 2x income? (Read 603 times)
Let's assume that Alice does 5 hashes per minute and Bob does 10 hashes per minute. Is Bob's income exactly 2 times higher in the long run?
Some basic math assuming that difficulty==10 (odds to find a nonce are 1/10 per attempt):
Odds to find a nonce after one attempt == 1 minus odds_to_fail == 1 - 0.9 == 0.1
Odds to find a nonce after two attempts == 1 minus odds_to_fail == 1 - 0.9*0.9 == 1 - 0.81 == 0.19
... three attempts ... 1 - 0.9*0.9*0.9 == 0.271
...
... five attempts ... 0.40951
...
... ten attempts ... 0.65132156
0.65 not equals 0.41*2, so I'm not sure that income ~ hashrate. What's wrong with my logical reasoning?
Some basic math assuming that difficulty==10 (odds to find a nonce are 1/10 per attempt):
Odds to find a nonce after one attempt == 1 minus odds_to_fail == 1 - 0.9 == 0.1
Odds to find a nonce after two attempts == 1 minus odds_to_fail == 1 - 0.9*0.9 == 1 - 0.81 == 0.19
... three attempts ... 1 - 0.9*0.9*0.9 == 0.271
...
... five attempts ... 0.40951
...
... ten attempts ... 0.65132156
0.65 not equals 0.41*2, so I'm not sure that income ~ hashrate. What's wrong with my logical reasoning?
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