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when observing the natural world many of us notice a somewhat beautiful dichotomy no two things are ever exactly alike but they all seem to follow some underlying form and Plato believed that the true forms of the universe were hidden from us through observation of the natural world we could merely acquire approximate knowledge of them they were hidden blueprints the pure forms were only accessible through abstract reasoning of philosophy and mathematics for example the circle he describes as that which has the distance from its circumference to its center everywhere equal yet we will never find a material manifestation of a perfect circle or a perfectly straight line though interestingly Plato speculated that after an uncountable number of years the universe will reach an ideal state returning to its perfect form this platonic focus on abstract pure forms remained popular for centuries and it wasn't until the sixteenth century when people try to embrace the messy variation in the real world and apply mathematics to tease out underlying patterns Bernoulli refined the idea of expectation he was focused on a method of accurately estimating the unknown probability of some event based on the number of times the event occurs in independent trials and he uses a simple example suppose that without your knowledge 3,000 light pebbles and 2,000 dark pebbles are hidden in an urn and that to determine the ratio of white versus black by experiment you draw one pebble after another with replacement and note how many times a white pebble is drawn this is why he went on to prove that the expected value of white versus black observations will converge on the actual ratio as the number of trials increases known as the weak law of large numbers and he concluded by saying if observations of all events be continued for the entire infinity it will be noticed that everything in the world is governed by precise ratios and a constant law of change this idea was quickly extended as it was noticed that not only did things converge on an expected average but the probability of variation away from averages also follow a familiar underlying shape or distribution a great example of this is Francis Galton's beam machine imagine each collision as a single independent event such as a coin flip and after 10 collisions or events the beam falls into a bucket representing the ratio of left versus right deflection or heads versus tails and this overall curvature known as the binomial distribution appears to be an ideal form as it kept appearing everywhere anytime you look at the variation of a large number of random trials it seems the average fate of these events is somehow predetermined known today as the central limit theorem but this was a dangerous philosophical idea to some and pavel nekrasov originally a theology by training later took up mathematics and was a strong proponent of the religious doctrine of free will he didn't like the idea of us having this predetermined statistical fate and he made a famous claim that independence is a necessary condition for the law of large numbers since independence just describes these toy examples using beans or dice where the outcome of previous events doesn't change the probability of the current or future events however as we all can relate most things in the physical world are clearly dependent on prior outcomes such as the chance of fire or Sun or even our life expectancy and when the probability of some event depends or is conditional on previous events we say they are dependent events or dependent variables so this claim angered another Russian mathematician Andrey Markov who maintained a very public animosity towards nekross off he goes on to say in a letter that this circumstance prompts me to explain in a series of articles that the law of large numbers can apply to dependent variables using a construction which he brags necros off and not even dream about so Markov extends Bernoulli's results to dependent variables using an ingenious construction imagine a coin flip which isn't independent but dependent on the previous outcome so it has a short term memory of one event and this can be visualized using a hypothetical machine which contains two types which we call States in one state we have a 50-50 mix of light versus dark beads while in the other state we have more dark versus light now one cup we can call state zero it represents a dark having previously in the other state we can call one it represents a light-being having previously occurred now to run our machine we simply start in a random state and make a selection and then we move to either state 0 or 1 depending on that event and based on the outcome of that selection we output either a 0 if it's dark or a 1 if it's light and with this 2 state machine we can identify four possible transitions if we are in state zero and a black occurs we loop back to the same state and select again if a light bead is selected we jump over to state 1 which can also loop back on itself or jump back to state zero if a dark is chosen the probability of a light versus dark selection is clearly not independent here since it depends on the previous outcome but Markov proved that as long as every state and the machine is reachable when you run these machines in a sequence they reach equilibrium that is no matter where you start once you begin the sequence the number of times you visit each state converges to some specific ratio or probability now this simple example disapproved necroses claim that only independent events could converge on predictable distributions but the concept of modeling sequences of random events using States and transitions between states became known as a Markov chain and one of the first and most famous applications of Markov chains was published by Claude Shannon you