The examples in documentation of MonadRandom does a good job on how monads are useful. I have it cut and pasted here:

There is some correspondence between notions in programming and in mathematics:

random generator ~ random variable / probabilistic experiment result of a random generator ~ outcome of a probabilistic experiment

Thus the signature

rx :: (MonadRandom m, Random a) => m a

can be considered as "rx is a random variable". In the do-notation the line

x <- rx

means that "x is an outcome of rx".

In a language without higher order functions and using a random generator "function" it is not possible to work with random variables, it is onlypossible to compute with outcomes, e.g.`rand()+rand()`

. In a language where random generators are implemented asobjects, computing with random variables is possible but still cumbersome [.]as you need to wrap the generators (Composition Pattern) in object and perform the computation as methods of the new object

In Haskell we have both options either computing with outcomes

do x <- rx

y <- ry

return (x+y)

or computing with random variables

liftM2 (+) rx ry

This means that liftM like functions convert ordinary arithmetic into random variable arithmetic [Since you can't do lift in OO, you would need to write your own version of each operation you like to support]. But there is also some arithmetic on random variables which can not be performed on outcomes [Or would be adhoc in OO]. For example, given a function that repeats an action until the result fulfills a certain property (I wonder if there is already something of this kind in the standard libraries)

untilM :: Monad m => (a -> Bool) -> m a -> m a

untilM p m = do x <- m

if p x then return x else untilM p m

we can suppress certain outcomes of an experiment. E.g. if

getRandomR (-10,10)is a uniformly distributed random variable between −10 and 10, then

untilM (0/=) (getRandomR (-10,10))is a random variable with a uniform distribution of {−10, …, −1, 1, …, 10}.

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