Bayes Theorem is a way to calculate conditional probabilities. This is important when you want to calculate how probabilities change when you see data. It is a way to measure how much you have learned. This is used in finance when we update our estimate of what a project is worth. When we see data, we update our estimate of probabilities and thus of expected cash flows. It is also used when we value projects using real options. With real options, we make a decision in the future based on information we have in the future but do not have today. We learn from this new information (usually) and thus it causes us to update our probability and expected cash flow estimates.
The settings to which we apply Bayes Theorem always have a similar structure. There are multiple possible states of the world (e.g. Berkeley or Cleveland). We have an initial guess of the probability of which state we are in (80% versus 20%). This initial probability guess is called the unconditional probability or the prior probability. You then observe data. In the real options lecture, this was the cash flow we observed or the results of the market research. We may learn from the data. If so, we update our estimated probability of which state we are in (Berkeley or Cleveland). The updated probability estimate is called the conditional probability or the posterior probability.
If you would like to review the logic of Bayes Theorem, I recommend a couple of Khan Academy Videos. Probability 7 and Probability 8 contain the most relevant material. They show the intuition and mechanics of Bayes Theorem. The concepts will also help you in other class assignments (oil drilling simulation homework) and life (when you need to update probabilities based on new data -- e.g. learning). Probability 7 and Probability 8 build on Probability 6. I don’t think you need to watch Probability 6, especially if you have read my description below. One feature I like in the way the material is presented in Probability 6 is the use of trees to describe the information. I find that decisions trees are a very useful tool for analyzing and explaining real options valuation.
In the Khan Academy videos, you will draw a coin from a bag of 10 coins. Nine of the coins are fair (heads comes up with 50% probability). One of the coins has heads on both sides. Which coin you have is the state of the world. You then flip the coin and it came up heads five times. This is the data you observe. You have an initial estimate of the probability that the coin you have is the two sided coin (10%). After you observe five heads (the data from which you learn), what is the revised (conditional or posterior) probability that the coin you have is the two sided coin? The difference between your initial probability estimate and your revised probability estimate is a measure of how much you have learned from the data. In a finance context, this will determine how much the expected cash flows and the value of a project (e.g. a stock price) changes.
Typo note: At about 4:40 of the Probability 7 video, the formula is incorrect, but the spoken words are correct. The formula should read P( 5/5 | N ) P( N ) = P( 5/5 AND N). This is noted in the comments. When there are typos in the Khan Academy sessions, these are usually noted in the comments.