An honest man is flipping a fair coin. After flipping ‘heads’ seven times in a row, a banker walks up and offers the honest man a bet of $50 if the next flip is ‘heads’. The banker hasn’t seen the previous seven flips. Should the honest man take the bet?
More importantly, do the previous flips effect the outcome of the next?
This is the concept of conditional independence. Clearly, the probability of any coin flip is 1/2, but the probability of eight flips all being the same is 1/256 (0.5^8). So, if the future depends on the past, can the future be determined within some finite probability?
To take the gambling metafore a bit further, slot machines are perfect examples of discrete random number generators. If you watch someone else play a slot machine for a while, the more they play and don’t win, the higher the probability that their text turn will be a jackpot. Ideally, a winning slot player should observe the other players in the casino and try to take over a ‘cold’ machine that has been played frequently but never produced a jackpot. Despite the social aspects of a slot machine or table ‘going cold’, these machines actually have the highest probability of making you rich.
Being able to predict the future under these very controlled situations doesn’t sound like a very useful super power, but it has many real world applications. Physicists use Markov Chain Monty Carlo (MCMC) models in determining the movement of subatomic particles, and it’s becoming popular with information theorists for behavioral analytics of internet click-streams.