Join Eddie Davila for an in-depth discussion in this video Bayes theorem, part of Statistics Foundations: Probability.
- Millionaires. Who are they, and how did they get their money? There's all sorts of data on this topic, but let's say you were given this set of data. Here's the amount of education millionaires in a certain country achieved. 20% had graduate degrees, 60% undergraduate degrees. 15% had only a high school diploma. And 5% did not even complete high school. Suppose for this same country, we also know how many inherited their wealth versus earning it. Here's the data. 40% of millionaires that are high school dropouts inherited their wealth. 20% of millionaires that had either a high school diploma or an undergraduate degree inherited their wealth. And only 15% of millionaires with graduate degrees inherited their wealth. What's the probability that a millionaire did not complete high school given that they earned their wealth? Well, let's set up a tree. Our first set of branches establishes the level of education. Our second set of branches then break up these different types of millionaires into those that inherited their wealth versus those that earned their wealth. By multiplying the value of the first branch times the value of the second branch, we can see the probability of each outcome. To simplify, let's say there are 1,000 millionaires. We can multiply each of our branch values times 1,000. We can now see how many millionaires we have in each category. For example, there are 200 millionaires with graduate degrees. 170 of them earned their money. 30 inherited their money. We can also see that by isolating only those that earned their money, 800 of our 1,000 millionaires earned their money. And of those 800, 30 did not graduate high school. That's 3.8%. So the probability that a millionaire did not complete high school given that the millionaire earned their money is 3.8%. Some of you may be wondering, "Is there another way to do this without the pictures? Is there a formula?" There is. Here is the formula for Bayes' theorem. Scary at first, but logical once you decode it. Probability of A given B. In this case, the probability of high school dropout given earned money. So our numerator is probability of dropout, 5%, times probability of dropout earns money, 60%. The denominator is adding up all the people that earned their money. First, for dropouts, they are 5% of the population. 60% of dropouts earned their money. So we multiply 5% and 60%. To that we add high school graduates. 15% of the population times 80% of them earned their money. For undergraduates, 60% of the population times 80% of them earned their money. And finally, for those with graduate degrees, 15% of the population times 85% of them earned their money. When we calculate all of this, we get an answer of 0.038 or 3.8%, the same thing we got using our probability trees. So whether you're looking at false positive data, crime data, educational data, science data, or even business data, Bayes' theorem can help you understand relationships and probabilities.
Eddie explains that probability is used to make decisions about future outcomes and to understand past outcomes. He covers permutations, combinations, and percentiles, and goes into how to describe and calculate them. Eddie introduces multiple event probabilities and discusses when to add and subtract probabilities. He describes probability trees, Bayes’ Theorem, binomials, and so much more. You can learn to understand your data, prove theories, and save valuable resources—all by understanding the numbers.