A Statistical Model of Bitcoin Mining

Bitcoin mining is often described with a simple calculation: estimate a machine’s revenue, subtract its electricity bill, and decide whether to keep it running. Our new paper “Expected Revenue, Risk, and Grid Impact of Bitcoin Mining: A Decision-Theoretic Perspective” (forthcoming in Energy Economics) argues that this leaves out something fundamental. Mining revenue is uncertain by design. A useful model must explain the average payoff, the risk of falling short, the chance of earning more than expected, and the choices miners make to manage that uncertainty.

The paper begins with proof-of-work itself. Each hash produced by a mining machine is effectively one lottery ticket. The machine runs a block header through Bitcoin’s SHA-256 function and checks whether the result falls below the network’s target. Almost every attempt fails; a successful one earns the block subsidy. Because Bitcoin’s mining difficulty sets the probability of success, we build mining economics directly from the protocol rather than relying only on historical revenue.

That is the paper’s first major contribution: a unified, forward-looking statistical model derived from first principles. Common measures such as “hash price” report realized revenue per unit of computing power at a particular time. They are useful snapshots, but they do not represent the uncertainty miners face beforehand. The new model calculates expected revenue, costs, and net profit for a given hash rate and operating period, linking them to electricity use and price, hardware efficiency, and Bitcoin’s price.

The paper calibrates the model using Bitcoin mainnet block 808,468 from September 19, 2023. Under its stated assumptions—an Antminer S19 at 110 terahashes per second, Bitcoin at $42,265, a 6.25 BTC block reward, continuous operation for one year, and electricity at $0.0885 per kilowatt-hour—the machine produces expected revenue of $3,724.03 at an electricity cost of $2,514.35. That generates income of about $1200 per machine per year, which gives a breakeven at about 3 years at 2023 S19 prices. These are not timeless profitability forecasts; they show that the model can be calibrated to real hardware and market data.

The second major contribution is to separate expected return from risk. Two operations can have similar expected revenue but very different chances of achieving it. One measure compares the volatility of mining revenue with its expected value. Another asks a practical question: how much computing capacity is needed to keep the probability of earning less than a chosen share of expected revenue below a specified limit?

Both approaches show that scale matters, but not one-for-one. Larger fleets make revenue more predictable, yet risk declines only at a square-root rate, so strict targets require disproportionately large fleets. In one calculation, a direct-mining facility using S19 machines for one year needs 28,171 machines for a 5 percent coefficient-of-variation target, compared with 3,130 machines for a 15 percent target. Under the paper’s assumptions, a 1,000-machine fleet also has a 35.32 percent probability of earning at least 110 percent of its expected annual revenue.

The third contribution is a framework for deciding how much capacity to place in a mining pool. Pools combine miners’ work and distribute payments under agreed rules, making income steadier. Fees and payout arrangements can reduce expected revenue. Our paper treats that reduction as an opportunity cost: pooling works like insurance, but it is not free.

Rather than framing the choice as “pool or no pool,” the model determines the minimum share of a fleet that must be pooled to meet a chosen risk limit. The rest can mine directly and preserve more upside. In one example using a payout inferred from Riot Platforms’ 2022 results, a 10,000-machine facility seeking at least 95 percent of expected return with no more than a 5 percent shortfall probability would allocate 2,191 machines to a pool. Partial pooling becomes a measurable decision rather than a rule of thumb.

These results matter beyond mining companies because financial risk management can change electricity demand. Bitcoin mines are flexible industrial loads that can reduce consumption quickly when prices rise or the grid is stressed. Yet identical facilities may respond differently if one has heavily pooled its production and the other relies on volatile direct-mining income. Our paper suggests that a highly pooled miner may operate across a broader range of electricity prices, while an unpooled miner may require a larger expected margin. Grid planners therefore need to understand how profitability, risk tolerance, fleet size, and pooling affect operating decisions.

The paper does not claim to capture every feature of the industry. Its current version simplifies transaction fees and pool payout rules and does not yet model location-specific factors such as congestion, carbon intensity, curtailment, or interconnection limits. Those are identified as areas for future work.

Its main achievement is a common mathematical foundation for studying mining economics and grid impact. By treating every hash as a probabilistic trial, the paper connects expected revenue, downside risk, upside potential, electricity cost, fleet size, and pool participation. For miners, that offers clearer planning tools. For grid operators, it offers a more realistic view of a large, flexible load whose behavior depends not just on average profitability, but also on uncertainty.