Bayes' Formula

Author's Note: This is a great video for teaching Bayes' Theorem in its simplest form.

Summary
In order to illustrate the utility of Bayes’ Theorem, the author draws upon two simple scenarios. First, suppose someone faces the decision of needing to choose between three doors. If the person making the decision does not have any prior knowledge about the situation, the scenario creates an unconditional probability. But, once the person receives new information about the scenario, the rational person should reconsider his/her decision and subsequent probabilities.

Bayes’ Theorem is about the introduction of new information used to adjust probabilities and create conditional probabilities. In the formula, P(G/U), P is the probability that G will occur, if U happens.

To illustrate the application of Bayes’ Theorem and conditional probabilities, the author illustrates a second scenario. Pretend that there is a 70% probability that the economy will grow and a 30% probability that the economy will slow (an unconditional probability). The author owns a stock that has an 80% chance of increasing if the economy grows. That same stock, however, only has a 30% chance of increasing if the economy slows. The 80% and the second 30% are conditional probabilities; they are based on the condition that the economy will grow or slow.

The author can then determine the scenario’s four conditional probabilities:
1) What is the probability that the economy will grow and the stock will increase?
2) What is the probability that the economy will grow and the stock will decrease?
3) What is the probability that the economy will slow and the stock will increase?
4) What is the probability that the economy will slow and the stock will decrease?

To answer these questions and determine their probabilities, the author uses the equation: P(UG) = P(U/G)P(G). Notice that this equation is longer than the first because this one incorporates two conditions: the economy will grow/slow and the stock will increase/decrease.

Decision Tree Analysis: Choosing Between Options By Projecting Likely Outcomes

MindTools.com

Summary
Decision Trees are useful for helping decision-makers to choose between several courses of action. Their structure allows decision-makers to explore their options and investigate the possible outcomes of those options. Additionally, Decision Trees help decision-makers to form a balanced picture of the risks and rewards associated with each possible course of action. To aid in the decision making process, Decision Trees provide a framework to quantify the values of outcomes with the probabilities of achieving them.

How To Make A Decision Tree:
1.) Draw a small square to represent a decision to be made on the left hand side of a large piece of paper, half way down the page.
2.) From this box, draw lines out towards the right for each possible solution, and write a short description of the solution along the line.
3.) At the end of each line, consider the potential/likely results. If the result taking that particular decision is uncertain, draw a small circle. If this results in another decision needing to be made, draw another square. Squares represent decisions and circles represent uncertain outcomes.
4.) Starting from the new decision square, draw lines out representing the potential options that could be selected. From the circles, draw lines representing possible outcomes.
5.) Repeat steps 2-4 until all possible decisions and outcomes are illustrated.

To begin the decision making process, start by assigning a score to each possible outcome. This score is an estimate of how beneficial the result is. Next, look at each circle and estimate the probability of each outcome.

Then, calculate the value of the uncertain outcomes (multiply the value of the outcomes by their probability). Start on the right hand side of the Decision Tree and work back towards the left. Only record the result to each respective square or circle from each set of calculations.

To evaluate each decision node, write down the cost of each option on each decision line. Then subtract the cost from the outcome value previously calculated. This value represents the benefit of that decision. Once all calculations are complete, choose the option that has the largest benefit (the final decision).