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Suppose a miner controls α fraction of the hashing power and the chain on which he is trying to mine has the mining rate given by f. How to find the probability of mining a block by the miner?

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In an ideal world, if a miner has α fraction of hashing power, the probability of them finding the next block is α.

It should not depend on any other factor, but in real-world network speeds, block propagation and verification speed complicate things.

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Probability

As MCCCS comented:

Your hashrate divided by total hashrate is the probability of you finding the next block.

As of March 2021

Item Hashrate MH/s
Typical desktop computer's i7 CPU 24
Antminer S19 Pro 110,000,000
Bitcoin network 156,000,000,000,000

So every ten minutes a CPU miner would have a 1 in 6,500,000,000,000 chance of earning 6.25 BTC. You have a 1 in 139,000,000 chance of winning the Euromillions lottery so lottery tickets will give you an overwhelmingly better return on investment in the longer term than does CPU mining. In both cases you are more likely to lose lots of money than see any profit.


"Expected earnings"

You can also key your hashrate into an online calculator to find your "expected earnings".

Setting Value Units
Difficulty Factor 1.54660989356e+13
Hash Rate 24.0 MH/s
Exchange Rate 50809.85 ($/BTC) [user]
BTC / Block 6.25000000

This Difficulty

Coins Dollars
per Day 0.00000000 BTC $0.00
per Week 0.00000000 BTC $0.00
per Month 0.00000001 BTC $0.00
this diff (est) 0.00000008 BTC $0.00

Next Difficulty [estimated]

Coins Dollars
per Day 0.00000000 BTC $0.00
\per Week 0.00000000 BTC $0.00
per Month 0.00000001 BTC $0.00

Of course, you might get lucky.


References

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