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Suppose there is mining pool XYZ. 10 miners 1 through 10 are subscribed to it and miners have varying hash power, some have 10 GPU mining rig while others have just a single CPU running 24 hours a day seven days a week. Suppose 12.5 BTC are won by XYZ each week.

Does it mean that each week one of those 10 miners win 12.5 BTC and XYZ then distributes BTC to each of the 10 miners in accordance with their contribution of hashing power?

OR XYZ gathers together hashing power of miners 1 to 10 to try to be the first to solve the puzzle to win 12.5 BTC ?

Is it possible for a solo miner with 10+ GPU rig to win 12.5 BTC block reward even though mining pool XYZ has much greater combined hashing power? I mean is there randomness also involved in wining block reward in addition to hashing power?

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OR XYZ gathers together hashing power of miners 1 to 10 to try to be the first to solve the puzzle to win 12.5 BTC ?

XYZ assigns work to miners in its pool. The work mines blocks that pay BTC to the pool's distribution account. When a block is mined, the pool distributes from the distribution account.

Is it possible for a solo miner with 10+ GPU rig to win 12.5 BTC block reward even though mining pool XYZ has much greater combined hashing power? I mean is there randomness also involved in wining block reward in addition to hashing power?

Yes. Everybody who is trying to mine a block has the same chance of succeeding with each hash they do. Mining a block is so hard that somebody somewhere succeeds on average once every ten minutes.

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