• How cpuminer searches nonces
• Why miners don’t use random nonces
• Your key question: does randomness help a slower miner?
• Important implication
• Where “randomness” actually appears
• Subtle edge case (for completeness)
• Bottom line
• 1. What you’re correct about
• 2. Where the reasoning breaks
  • The critical fact:
• 3. Sequential vs random search
• 4. The key probability result
• 5. Your “lucky starting point” idea
• 6. The real advantage mechanism
  • 1. More hashes (dominant factor)
  • 2. Different search space (avoiding overlap)
• 7. Important subtlety (your intuition refined)
• 8. Final conclusion
  • What the gambler’s fallacy is
  • What you’re actually thinking
  • Why it feels like it should help
  • Why it doesn’t actually help
  • The deeper concept
  • Analogy (more accurate than gambler’s fallacy)
  • Final classification
  • Bottom line
  • The key point
  • Why your intuition still feels right
  • Important clarification
  • What actually matters for “lottery success”
  • If you still want to randomize
  • The honest takeaway
  • Bottom line
  • 1. Probability per hash
  • 2. Your hash rate
  • 3. Probability per second
• Final answer (your odds)
  • Per second:
  • More intuitive time scales
• Interpretation
  • The paradox you’re noticing
  • Intuition framing
  • What makes it feel different from normal lotteries
  • The honest conclusion
  • 1. Uniform randomness over huge spaces
  • 2. Law of large numbers vs human scale
  • 3. Memoryless processes
  • 4. Expectation vs possibility
  • 5. Why mining feels uniquely interesting
  • Bottom line

“I have a technical question about cpu-miner/minerd: How does it rotate nonces? Is it random or serially? Assuming that a powerful miner and a slow miner happen to be mining the same block template, i.e. transaction set, then the only way for the slow miner to have an advantage is if they’re rotating nonces randomly and therefore have a slim chance to randomly arrive at the correct nonce first.”

Short answer: it’s effectively serial (deterministic), not random, and that does not give a slower miner any advantage.

§How cpuminer searches nonces

In Bitcoin mining:

• The block header has a 32-bit nonce field
• Miners try different values until the hash is below target

cpuminer (and essentially all miners) do:

• Increment nonces sequentially (nonce++)

• Each thread works on a different slice of the nonce space

• When nonce space is exhausted, they:

  • update extraNonce in the coinbase (changes merkle root)
  • or receive a new job from Stratum (new template)

So the search space looks like:

Thread 0: nonce 0 → 1 → 2 → ...
Thread 1: nonce offset → offset+1 → ...
...

Not random sampling.

§Why miners don’t use random nonces

Because:

• The hash function behaves like a random oracle

• Sequential search is already equivalent to random sampling over the space

• Randomizing would:

  • add overhead
  • risk duplicate work across threads

§Your key question: does randomness help a slower miner?

No.

Even if two miners:

• have the same block template
• explore different nonce orders (random vs sequential)

The probability model is:

• Each hash attempt is an independent Bernoulli trial
• Success probability per hash = extremely small constant

So:

Probability of success ∝ number of hashes performed

A faster miner simply performs more trials per second.

§Important implication

There is no “lucky ordering advantage”:

• The correct nonce is not “special” in position
• It is just one of many valid hashes
• There are typically many valid nonces, not just one

So:

• Faster miner → more hashes → higher probability
• Slower miner → fewer hashes → lower probability

§Where “randomness” actually appears

Randomness enters in:

1. Hash output itself (SHA256d behaves pseudo-random)
2. extraNonce changes
3. Template updates from mempool / new blocks

Not in nonce selection strategy.

§Subtle edge case (for completeness)

If two miners:

• use identical template
• start at identical nonce
• iterate identically

They could duplicate work.

That’s why pools/Stratum assign:

• different extraNonce ranges
• or different work units

But for solo mining, this isn’t an issue.

§Bottom line

• cpuminer uses sequential nonce iteration
• Randomizing nonce order does not improve odds
• Mining success is purely proportional to hashrate × time

Your intuition about “random chance advantage” is understandable, but in practice the math cancels that out completely.

“The difference is to be working on a different set than all other miners. A certain set of data will produce a certain hash that appears random, but the same set of data always produces the same hash. The goal is to arrive at a set of data that produces a hash with a low enough value before anyone else. What you certainly do not want to do as a slow miner is be working on the exact same set of work as a faster miner or they will be mathematically guaranteed to arrive at the correct set of data to produce the valid hash first. It is mathematically unlikely but possible to randomly start at a starting point where you run through very few iterations before arriving at a valid hash. For a lottery miner, some greater degree of randomness can put luck on your side.”

You’re right about one part and slightly off on the conclusion. The nuance matters.

§1. What you’re correct about

“You don’t want to be working on the exact same set of work as a faster miner”

Correct. Avoiding duplicate work is important.

That’s exactly why mining systems use:

• different extraNonce values (coinbase)
• different nonce ranges per thread
• different block templates across the network

So each miner explores a different region of the search space.

§2. Where the reasoning breaks

“Randomness can give a slow miner an advantage”

This is the key misconception.

§The critical fact:

For a given block template, the mapping:

input (nonce, extraNonce, header fields) → hash

behaves like a uniform random function.

So:

• The location of valid hashes in the space is uniformly random
• There is no clustering near certain nonce values
• There is no “better starting point”

§3. Sequential vs random search

You’re comparing:

• Sequential search:

  0, 1, 2, 3, ...
  

• Random search:

  837261, 12, 991827, ...
  

But mathematically, both are equivalent:

Sampling without replacement (sequential) ≈ sampling with replacement (random)

Because:

• The valid solutions are randomly distributed
• Every trial is independent

§4. The key probability result

Let:

• p = probability a single hash meets target
• n = number of hashes you compute

Then:

P(success) = 1 - (1 - p)^n

This depends only on n, not on:

• order
• randomness
• starting point

§5. Your “lucky starting point” idea

“You might randomly start near a valid nonce”

That’s true in a single trial sense, but:

• Sequential miner has the same probability of hitting that solution in its first k attempts
• Random miner does not increase probability, it just changes ordering

There is no bias to exploit.

§6. The real advantage mechanism

There are only two real advantages in mining:

§1. More hashes (dominant factor)

• Faster miner → more trials → higher probability

§2. Different search space (avoiding overlap)

• Ensures you’re not wasting effort duplicating work

But:

Randomizing order does NOT create a third advantage

§7. Important subtlety (your intuition refined)

Your instinct is actually pointing to something real:

• You want to avoid correlated work
• You want to maximize coverage of the search space

But that is achieved by:

• partitioning nonce ranges
• varying extraNonce
• using independent templates

—not by randomizing nonce order.

§8. Final conclusion

• cpuminer’s sequential nonce scanning is optimal
• Random nonce selection does not improve lottery odds
• The only thing that matters is:

how many unique hashes you compute before someone else finds a block

Your “lottery intuition” is natural, but Bitcoin mining is not a lottery with hidden bias—it’s a pure uniform probability process over a massive search space.

“So essentially what I am arguing is a gambler’s fallacy”

Not exactly. It’s closer to a gambler’s fallacy, but more precisely it’s a misapplied intuition about search order in a uniform random space.

§What the gambler’s fallacy is

The classic gambler’s fallacy is:

Believing past outcomes affect future probabilities (e.g., “after many reds, black is due”)

That’s not what you’re arguing.

§What you’re actually thinking

Your idea is:

“If I randomize my search, I might get lucky and hit a valid nonce early.”

That’s not about past outcomes—it’s about search strategy.

§Why it feels like it should help

You’re imagining:

• The solution is somewhere in a huge space
• If you “jump around randomly,” you might land near it quickly
• Sequential search feels “slow” or “plodding”

This is a very natural intuition.

§Why it doesn’t actually help

Because:

• The valid hashes are uniformly randomly distributed
• There is no “region” that is more likely to contain a solution
• There is no “closer” or “farther” in hash space

So:

• Sequential search = systematic sampling
• Random search = random sampling

Both give the same probability after N tries.

§The deeper concept

What this really is:

Confusing “random order might get lucky early” with “random order increases probability”

It doesn’t increase probability—it just reorders when luck might occur, without changing how likely it is overall.

§Analogy (more accurate than gambler’s fallacy)

Imagine:

• A deck of 4 billion cards
• A few are winners
• You draw cards one by one

Two strategies:

1. Go through the deck in order
2. Shuffle and draw randomly

Your chance of finding a winning card after N draws is the same.

§Final classification

Your intuition is best described as:

• Not classic gambler’s fallacy

• Not wrong reasoning, just misapplied randomness intuition

• Specifically:

  Belief that random sampling improves success probability in a uniform space

§Bottom line

• Randomizing nonce order can change when you get lucky
• It does not change how likely you are to get lucky
• Mining success remains strictly proportional to hashes performed

““Randomizing nonce order can change when you get lucky” Intuitively, I want to do this“

That intuition is understandable—but in this case it doesn’t give you any real advantage, even for a “lottery” mindset.

§The key point

You don’t control when you get lucky by choosing an order.

Because:

• The mapping from input → hash is effectively random
• The “winning” hashes are randomly scattered across the space
• There is no concept of “closer to a solution”

So while it feels like:

“Random order might hit a winner earlier”

In reality:

Any order is equally likely to hit a winner at any point

§Why your intuition still feels right

You’re thinking in terms of:

• “What if I just happen to start near a winning nonce?”

That’s valid for a single run, but:

• A sequential miner has the exact same chance that its first few tries include a winning nonce
• Randomizing doesn’t bias that probability—it just reshuffles outcomes

§Important clarification

“Randomizing can change when you get lucky”

This statement is true only in hindsight, not as a strategy.

• Yes, if you replay the same run with a different order, the winning hit might appear earlier or later
• But before you start, all orders have identical probability distributions

So you can’t engineer earlier luck

§What actually matters for “lottery success”

Only two things:

1. How many hashes you compute
2. Whether your work overlaps with others

You already solved #2 with:

• your own node
• your own ckpool instance
• unique extraNonce space

That’s the real “independence” factor.

§If you still want to randomize

You can—but it will:

• Not increase probability
• Slightly reduce efficiency (extra overhead)
• Potentially duplicate work across threads if done poorly

So it’s mostly cosmetic.

§The honest takeaway

Your instinct is basically:

“I want to feel like I have a shot at getting lucky early”

That’s completely reasonable psychologically.

But mathematically:

You already have that exact same chance without doing anything extra

§Bottom line

• You cannot bias luck timing through nonce order
• You already have a fair “lottery ticket”
• More hashes = more tickets
• Order of tickets = irrelevant

“Based on the hashes per second I showed you earlier, calculate the odds as a ratio to one.”

Use the network difficulty from your ckpool log:

difficulty ≈ 135,594,876,535,256

§1. Probability per hash

For Bitcoin:

P(success per hash) ≈ 1 / (difficulty × 2^32)

So:

≈ 1 / (1.3559e14 × 4.29e9)
≈ 1 / 5.82e23

§2. Your hash rate

≈ 6.7e7 hashes/second (≈ 67 MH/s)

§3. Probability per second

P(success per second) = hashrate / total space
≈ 6.7e7 / 5.82e23
≈ 1 / 8.7e15

§Final answer (your odds)

§Per second:

≈ 1 in 8,700,000,000,000,000
(1 in 8.7 quadrillion)

§More intuitive time scales

Per day:

8.7e15 / 86,400 ≈ 1 in 100,000,000,000
(1 in 100 billion per day)

Expected time to hit one block:

≈ 8.7e15 seconds
≈ 275 million years

§Interpretation

• Every hash is a “lottery ticket”
• You’re generating ~67 million tickets per second
• The network requires ~5.8 × 10²³ tickets on average to win

So your setup is:

Extremely low probability, but non-zero

Which is exactly what makes it a true lottery miner.

“275 million years… but it COULD happen any second now haha”

Yes — and that statement is exactly correct in the strict probabilistic sense.

Each hash attempt is independent, so:

P(success in next second) is always the same

No matter whether you’ve been mining for:

• 1 second
• 1 hour
• 1 year

There is no memory in the process.

§The paradox you’re noticing

Two statements that both hold:

1. Expected time: ~275 million years
2. Could happen next second: also true

This is not a contradiction because:

• The distribution is geometric (memoryless)
• Most probability mass is spread over an enormous tail
• But there is always a tiny probability spike at “now”

§Intuition framing

Think of it like:

Every second:
You roll a die with ~8.7 quadrillion sides
You need to hit exactly one number

• You are overwhelmingly likely to not hit it for a very long time

• But there is always a valid outcome where:

  you hit it on the next roll

§What makes it feel different from normal lotteries

In a normal lottery:

• You buy one ticket → fixed chance

In mining:

• You are continuously buying tickets:

  • ~67 million per second

So psychologically it feels like:

“I’m constantly in the game”

Which you are—but the odds per unit time remain extremely small.

§The honest conclusion

• You are participating in a real, fair lottery
• Your chance right now is non-zero
• But the expectation is astronomically long

So:

Yes — it could happen any second
But it almost certainly won’t

Both statements are simultaneously true without contradiction.

“probability math and randomness are amazing”

They are—especially in cases like this where intuition and reality diverge so sharply.

What you’re seeing is the combination of a few powerful ideas:

§1. Uniform randomness over huge spaces

SHA256d behaves like a random function, so:

• No patterns
• No shortcuts
• No “better regions”

Every hash is equally meaningless—and equally valid as a lottery ticket.

§2. Law of large numbers vs human scale

• The network operates at exahash scale
• Your miner operates at megahash scale

Both are doing the same thing, just at vastly different scales. The math doesn’t change—only the magnitude.

§3. Memoryless processes

This is the part that feels the strangest:

• The past doesn’t matter
• The next attempt is always a fresh roll

That’s why:

“275 million years expected” and “could happen right now”

can both be true.

§4. Expectation vs possibility

Expectation describes the average over many universes. But you only live in one timeline.

So while almost all timelines look like:

nothing happens for millions of years

there exist rare timelines where:

you hit a valid block almost immediately

§5. Why mining feels uniquely interesting

Bitcoin turns pure math into something physical:

• Electricity → hashes
• Hashes → probability
• Probability → real-world payout

It’s one of the cleanest real-world examples of applied probability you can interact with directly.

§Bottom line

You’ve basically built a machine that continuously samples from an astronomically large random space and occasionally checks:

“Did I win?”

That’s why it feels so compelling.