How many times have you seen a TV talk show mention how someone failed a polygraph, so they must have cheated! Lie detectors don’t lie, right? How many times have you had a friend get bad news on a medical test and see that emotional toll that took on them?
What could these two have in common? They could both be helped by Bayes’ theorem.
Bayes’ theorem is an idea in statistics that allows us to update the probability of something happening based on new evidence. You can see what it looks like in the featured image above. However, the best explanation I have ever seen comes from this image:
Let’s go through each part of this image individually:
- H: This is our hypothesis, or what we think is true. In the talk show example, this could be “I think my man is cheating on me.”
- D: This is our new information. For the talk show, it’d be the results of our polygraph test.
- P(H|D): This is the likelihood that the hypothesis is true given our new information. “What are the odds that my man is lying about cheating on me if the polygraph says he’s a liar?”
- P(D|H): This is the odds of seeing the new information if the hypothesis is true. Here, it’d be the odds that the polygraph would say someone was lying when they were lying. This turns out to be 88% (Rice, 2007).
- P(H): This was the probability that the hypothesis was true before the new information. For example, 22% of men say that they have cheated on a significant other.
- P(D): This the odds of the test saying the person was lying in every outcome. Here, you’d use the 88% it says someone is lying when they are, and multiply the odds that your man is lying (22%). You’d also have to add the odds the test says they are lying when they are not (false positive), about 14% (Rice, 2007), multiplied by the 78% chance your man isn’t a cheater.
This gives us the following result:
,
or 63.9%. This means that less than two out of three of those polygraph results are actually accurate. This low success rate, by the way, is one of the reasons most states don’t allow polygraphs in court.
But what about the bad medical test? How could it help us there? In statistics circles, the common example is the case of mammogram result that comes back positive for cancer. For our purposes, we’ll use numbers used by The New Yorker when they talked about this problem in this article from 2013. The numbers they quote are as follows:
- The chances that a women in her forties has breast cancer is 1.4% (meaning 98.6% don’t).
- The chances that a mammogram comes back positive when the woman has cancer is 75%.
- The chances that the mammogram comes back positive when the woman doesn’t have cancer is 10%.
Let’s imagine your friend in her 40s gets a positive result back on a mammogram. What are the odds that she actually has cancer? Bayes’ theorem tells us the probability would be
,
or only 9.6%. If that seems low to you, don’t worry: one study showed that 95% of physicians given similar numbers incorrectly estimated the likelihood to be about 75% (Rice, 2007). Your friend should be concerned and have further work done, but you can comfort her by saying that there is an over 90% chance it is a false positive for her.
What can Bayes’ do for you? It can help make talk shows more laughable, and it can help bring comfort when tests go south. It helps to give perspective and a more realistic view of the world. As it turns out, it’s pretty useful.
We’ll continue to explore these ideas in movie night this week with some videos that give even more examples about how Bayes’ theorem can help you every day.
The odds that those will be entertaining? 100%.
When You Will Use This 
Garin,
This is an awesome post! I loved Statistics in high school…it is probably the only math class I actually enjoyed besides Algebra I my freshman year. I have never heard of Bayes theorm but what an interesting concept. I knew lie detectors were far from perfect and now I know why they don’t hold up in court. I feel like I actually learned something today!
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I’m glad that you found it educational! For the polygraph example, we assumed that more than 1 in 5 men were cheaters. The accuracy gets even worse when you consider routine polygraph tests that some employers used to use in job hunts. The rate of false positives jumps from about 35% to over 90%. Nowadays, they are most often used by police to intimidate suspects who think polygraphs work like on TV and trick them into confessing. Still, it makes for more dramatic daytime television!
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This is great! I wish I would have been taught math with such applicable examples. I might remember more of it! Coincidentally, I’m reading a book on Hashimoto’s disease written by an expert who repeatedly comes back to the fact that so many people are mis- or undiagnosed when it comes to thyroid issues, because the test results are misleading, and most mainstream doctors often consider the test results directly, without considering the errors in the tests (they forget to treat the person, because they’re focused on the numbers – which they don’t question, incidentally). Interesting!
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A fun aspect of Bayes’ theorem is that you can adjust the prior based on the information of the current case. So if the patient has symptoms that are extra concerning, you may want to increase the prior from what it was. If the disease affects only 1% of the population, but you know that a quarter of the people you’ve seen with these symptoms have the disease, there’s nothing wrong with adjusting the prior to 25% to get a more accurate result. Bayes’ isn’t a hard-and-fast mathematical formula as much as a useful way to update expectations from what they were before the test (the prior) to after the test (the posterior). A great example is used in this video by Veritasium where just adjusting the prior from 1% to 9% can have a big difference, depending on the problem. Bayes’ theorem can help doctors avoid being too focused on the test, just like your book warns about! Thanks for sharing!
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I found this post really interesting. I was shocked to find out a polygraph is accurate less than two out of three times. I always see them used on crime shows to convict or clear people. Guess you shouldn’t trust everything on tv… 🙂
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Sad, but true! The test was only as “accurate” as shown here because we assumed that every man that had an affair would lie about it. If you were to include the people who have affairs and fess up when caught, the test would actually be less accurate than shown here. If only 64% of the men who affairs proceeded to lie about it when confronted, the test drops down to an accuracy of 50%. Failed by TV again!
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I think this is a great way to teach math and make it relatable! I don’t know of any kids that don’t stay home sick and watch Maury to see people fail lie detector and paternity tests. I also like that you will apply this to movies and find more ways to connect it! Those connections are what keeps learning in the mind!
Thank you for sharing!
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The more people can relate to math, the more they care about it, the more they study it, the more they understand it. I think English teachers have been doing that for years. English teachers take the readings and show how they relate to modern-day life. If math teachers did the same, I think fewer students would ask why they have to take it.
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What a clever post, Garin. You brought Bayes’ theorem into common conversation by hooking the reader with what is known and then moving it into the unknown by applying the the theorem to situations which the reader would know to show how it works. Love it! Thanks for sharing!
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Thank you! The more people see math as a way to relate to their world instead of useless tasks best left to “nerds” and calculators, the less people will see math as an insurmountable challenge! At least, that’s the hope. Thanks for your comment!
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