How Mining Pool reduces the variance in mining rewards ?

There are many advantages of joining the mining pools. For now, i will just only focus on increasing the overall hash rate percentage.

Bernaulli’s distribution states that for every discrete distribution which has two outcomes “success” and “failure”, if the probability of success is p (where 0<p<1) then probability of failure is (1-p).

Lets assume a miner controls fraction “p” of overall mining power and has prob. p to mine the new block then (1-p) prob. of not mining the new block.

so according to Bernaulli’s distribution, variance = p*(1-p).

so How many blocks are mined in a year ?

every 10 min = 1 blocks

one hour = 6 blocks

one day = 24*6 blocks

one year = 365246 blocks

we find that the miner who has prob. p of mining the new block is expected to mine 365246*p blocks per year.

then variance = 365246p(1-p).

Now we will calculate the standard deviation (in blockchain, it is the deviation that how much this miner’s reward is going to be deviated from the expected reward). I hope you know about the reward system for miners.

standard Dev. = [sqrt]Variance / expected return

variance = 365246p(1-p)

expected return = 365246*p

so std. dev = [sqrt]365246p(1-p) / 365246*p

now suppose a miner owns 0.01% of hash rate in network i.e prob. p=0.0001

std. dev = 0.4364 or 43.64 % deviation for hash rate 0.01%. (by putting the value in the above formula)

The solution to decrease this deviation and variance is to join mining pools and work together to increase the overall hash rate percentage, which is exactly working now a days.