The relationship between arbitrage opportunity and conditional probability appears to be so straightforward that it may seem dull to dig any deeper; but I consider it interesting and quite revealing.
What is arbitrage? The concept of arbitrage is very simple: profit from buying something at one price and selling it simultaneously at a different price. The key is that you are capitalizing on a price differential. This profit is sometimes called a free lunch or risk-less profit. We see arbitrage every day and probably never even recognize it.
What is conditional probability? It sounds intimidating but it is actually simple and straightforward. Probability is simply the chances of a given event occurring out of a possible set of outcomes. Note that possible is an important word here. If I flip a coin, I know that the only two possible outcomes are head or tails. Conditional probability means that the chances of one event occurring depend on the chances of a prior event happening. For example, if I have a deck of cards and pull two face cards out, then I know that the chances of me pulling another face card have changed.
Going back to the relationship, think about it. It is just something to ponder. Arbitrage is the act of using information that the market has not yet adjusted for and profiting from it. If people are buying King Kong DVDs on EBay for $25 and I know that I can buy them at Wal-Mart for $15, then I will exploit that arbitrage opportunity and make a risk-less profit of $10 per DVD. Not bad. Probability, as does arbitrage, relies on the availability of information. The probability of an outcome occurring, as is the ability to make an arbitrage profit, is dependent upon what we know. As information changes, so does the probability.