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Ethereum: How is the probability of winning a block calculated from the difficulty?

Published by Terlumun Aondover on February 13th, 2025.

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Understanding the probability of winning a block in Ethereum

The Ethereum network uses a Proof-of-Work (PoW) consensus algorithm, which requires miners to solve complex mathematical puzzles to validate transactions and create new blocks. The probability of winning a block is determined by the difficulty level of the puzzle.

In this article, we will delve deeper into how the probability of winning a block is calculated in Ethereum, including the Bitcoin wiki page that mentions it.

The Difficulty Formula

Ethereum: How is the probability of winning a block calculated from the difficulty?

The difficulty formula in PoW is based on the hash rate of the network’s mining pool and the current time. The formula can be simplified as follows:


Difficulty = (hash rate * time) / 2^32

This formula takes into account the total number of hashes per second (hash rate), the current number of blocks, and the desired difficulty level.

The relationship between block difficulty and reward

The Bitcoin wiki page mentions that the probability of winning a block is inversely proportional to the block reward. In other words, as the block reward increases, the chance of winning a block decreases. This relationship can be expressed mathematically:


P(win) = (1 / block_reward)

Probability Calculation

To calculate the probability of winning a block, we need to know the current block reward and the network difficulty level. The Bitcoin wiki page on Ethereum provides the following formula to calculate the block reward in Ethereum:

block_reward = 10^18 * (1 / difficulty)

Using this formula, we can plug in values to calculate the probability of winning a block.


block_reward = 10^18 * (1 / 4000000)
block_reward ≈ 2.5e-8
Difficulty = (hashrate * time) / 2^32

Example calculation

Suppose we know that the hash rate is about 100 TH/s and the current block count is “n”. We can use the above formulas to calculate the probability of winning a block:


block_reward = 10^18 * (1 / difficulty)
block_reward ≈ 2.5e-8

Assuming a block reward of 10^-9 satoshis per block
block_reward_per_block = 10^-9
Difficulty = (hashrate * time) / 2^32
difficulty ≈ 4000000
Block_Winning_Probability = block_reward_per_block / difficulty
block_winning_probability ≈ 1.25e-4

Conclusion

In conclusion, the probability of winning a block in Ethereum is calculated using the Bitcoin wiki page as follows:

Difficulty level affects the number of hashes possible needed to solve the puzzle.

The inverse relationship between block difficulty and block reward means that as the block reward increases, the chances of winning a block decrease.

By understanding this formula, we can better understand the underlying mechanics of PoW on Ethereum and make more informed decisions about our investments or participation in the network.