Join Eddie Davila for an in-depth discussion in this video Using probability trees, part of Statistics Foundations: Probability.
- Disease testing, always nerve wracking for the patient. The hope is for negative results. But unfortunately some patients will receive positive results indicating disease. It's very scary. But a doctor will likely retest in the hopes that there was an error. The hope that this first test was a false positive. So what's the probability that this first test might be wrong. First, we need to realize that we are considering two different events. Event one disease or no disease. Event two test positive or tests negative. Next, we need some statistics. According to doctors, only one in 10,000 people has the disease. This is helpful in evaluating event one. Now for the stats on event two, the testing company states that those with the disease will test positive 99% of the time. But 1% of those with the disease will test negative. That's a false negative also called a type two error. How about the uninfected? Well, 2% of uninfected patients will still test positive for the disease. In other words, 2% of healthy patients will get a false positive also called a type one error. To solve this, let's use our probability trees. Event one, does the patient have the disease? One person does, 9,999 do not. So our disease branch has the value of 0.0001. Our healthy branch has a value of 0.9999. Then we can move on to event number two, did the patient test positive? For those that actually have the disease, 99% test positive and 1% of patients with the actual disease will test negative. For those patients that do not have the disease, 98% will test negative 0.98 but 2% will test positive 0.02. These are the folks that get false positives. Let's calculate the value of each branch. The value of disease patients that test positive is 0.000099. The value of disease patients that test negative is 0.0000001. The value of healthy patients that test negative is 0.979902. The value of healthy patients that test positive is 0.019998. What do all of these decimals mean? Well, if 1 million people are tested for the disease, 100 people will have the disease. The test will catch 99 of those positives but one person with the disease will actually test negative. Let's move on to the other side of the tree. Remember 100 have the disease. This means that 999,900 are healthy, they do not have the disease. 979,902 of these people will test negative but 19,998 of these healthy people will get a positive result. So, finally, back to our question, if someone tests positive, what is the probability they actually have the disease? Well out of 1 million people, 20,097 will test positive but only 99 of those people will actually have the disease. So if you test positive, there is a 0.5% chance you actually have the disease. Only one out of every 200 people that test positive for the disease actually has the disease. That's got to make that patient feel at least a little bit better. What we did here is the basis for what we call Bayes theorem. It's not only interesting, it's actually very useful. So let's look at one more problem and get a better look at the mechanics of what the Reverend Thomas Bayes did with the use of conditional probability.
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.