It seems kind of important that a doctor can correctly interpret the following scenario:

- A 40-year old woman has a positive mammography in a routine screening.

The doctor is told the following about scanning for breast cancer:

- 1% of women at age forty who participate in routine screening have breast cancer.
- 80% of women with breast cancer will get positive mammographies (which means there are 20% false negative).
- 9.6% of women without breast cancer will also get positive mammographies (known as a 9.6% false positive)

What is the probability that she actually has breast cancer?

Only 15% of the doctors surveyed estimated the correct probability; most doctors estimated the probability to be between 70% and 80%, which is wildly incorrect.

That’s kind of scary.

This is a classical example of where Bayes’ Theorem should be used.

The problem, according to an editor at **Read the Sequences**, is that Bayesian reasoning is very *counterintuitive*. People do not employ Bayesian reasoning intuitively, find it very difficult to learn Bayesian reasoning when tutored, and rapidly forget Bayesian methods once the tutoring is over. This holds equally true for novice students and highly trained professionals in a field.

Here is **Bayes Theorem**:

**OR:**

Clear as mud, right?

No wonder people claim that it is not intuitive.

But let’s try to apply it to the mammography screening.

We are trying to solve for the probability of having cancer, given a positive screening. This is **P(A|B).**

Here is what the other terms mean:

**P(A)**= the overall probability of cancer; in this example, this probability is one percent, or**.01****P(B|A)**= probability of positive screening, given that you have cancer; in this example, this probability is 80 percent, or .**80****P(B)**= the overall probability of getting a positive result; this is the most challenging number to calculate.

To calculate the **P(B)**, let’s reword the original scenario.

Let’s assume that 1,000 women of age 40 take the mammography screening. Of these 1,000 women, 1% have cancer, which is 10 women. Of these 10 women, the mammography test will give the correct result for 80% of them, or **8 women**. Of the 990 women in this group who do not have cancer, the mammography test will falsely say that 9.6% of the women do have cancer, which is approximately **95 women**.

Thus, in total, out of the 1,000 women who take the mammography test, **103 (8 + 95)** will test positive, or about a **10.3 percent** chance, or .**103** **(103/1000).** **This is P(B)**.

Let’s put it all together now, using the first formula in the blue box above:

**P(B|A) * P(A) = =.8*.01 = .008 (in English, these are the odds of testing positive, given that you have cancer times the chance of having cancer)**

**P(B) = .103 (from above) (in English, this is the probability of testing positive)**

Dividing one by the other, we get **P(A|B) = ** **.008/.103 = .078, or 7.8%**

**We could have also calculated the denominator using the alternative formula shown above. We already know that**

**P(B|A) * P(A) = =.8*.01 = .008 these are the odds of testing positive given that you have cancer times the chance of having cancer**

**P(B|not A) * P(not A) = .096 * .99 = .095 these are the odds of testing positive given that you do not have cancer times the chance you do not have cancer.**

**Thus,**

**P(B|A) * P(A) plus ****P(B|not A) * P(not A) = .008 + .095 = .103 ****(in English, this is the probability of testing positive)**

So to answer the original question: **if a woman tests positive for cancer, there is only a 7.8% chance that she has cancer**; **dramatically different than the 70 to 80% probability that doctors had estimated.**

Perhaps an easier way to think of this would be to go back to the example of 1,000 women.

We have shown **103** of these women will test positive; however, only **8** of them will have cancer, **95** will not have cancer. Thus the probability of having cancer, given that you tested positive is **8/103, or 7.8%**.

As noted above, this may not be an intuitive result.

When you are told that 80% of women with cancer will test positive, that is not the same question as asking that if a woman tests positive, what are her chances of having cancer.

Since there are so many women more women who do not have breast cancer, and some of those women will test positive, that creates a much larger population of women who test positive and don’t have cancer (95) compared to the number who test positive and have cancer (8).

In conclusion, if you made it this far, then:

- you are a stat nerd like me (less than one percent chance)
- you need to find better reading material (greater than 99 percent chance)

I also wonder, given that a person has read this far, what the odds are that they will hit the like button…

- top image from
**LinkedIn**

Maybe this is why it’s a good idea to get a second opinion. But don’t go to a doctor for your second opinion, go to a mathematician.

LikeLiked by 5 people

or maybe every doctor’s office should have a stat person working there…

LikeLiked by 1 person

And maybe this is why you sometimes hear on the intercoms at hospitals, “Dr. So-and-so to the ER, stat!”

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🙂 I never thought of that; who knew stat people were in such demand!

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I pushed “like” even if I kind of skipped the numbers part. LOL!

I agree that its scary about those Drs giving such a crazy high answer! I don’t do Math but I even knew the percent was much lower.

I also was thinking of how ironic on this timing of your post. I go soon for my appt to test all those percentages.

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all I wanted was the like anyway 🙂

good luck with you test!

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And all I want is the money you offered to pay your readers. LOL! 😉

Thanks!

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once I monetize my blog, I will share the wealth… 🙂

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Sounds like a deal. Like sometime in the next century? 😄

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you’re being a little optimistic… 🙂

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😄

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I don’t expect my Doctor to understand statistics or mathematics. It’s why our Doctors can plug data into an algorithm and get the answers they need. That’s a better tool for them. Same as I don’t expect a statistician to know what the signs of breast cancer are.

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hopefully doctors are taking the time to do as you suggest…

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Hopefully, competent radiologists and oncologists who deal with breast cancer will have a good idea (intuitively) of the likelihood that a 40-year-old woman with a positive mammography actual has breast cancer without performing a mathematical calculation and will use additional tests and analysis before announcing a diagnosis. I wonder if the doctors surveyed were radiologists and oncologists who deal with breast cancer. If they were, that’s very concerning. Even so, it is surprising that MDs may be as bad at math as I am.

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you would hope they are versed in such matters; but this was just one example of how many people, even very bright people, do not understand some key statistical principles.

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Yep. I get your point. MDs at one time were even good at mathematical principles. If you asked lawyers a similar question, the percentage who could construct the right equation and get the right answer would be about 1%.

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I think many times, at least when I was in school, people who went on to med school were usually the smartest people in the class. I don’t think I could say the same thing about those who went to law school…

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I actually didn’t mind my college Stat classes when I was studying for my BSN. I preferred my statistical application class over the actual general stats though!

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I think the application-oriented courses are always more interesting!

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I used to think math was cool. Are you trying to change my opinion?

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You should see how many people I turn off from accounting! 🙂

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Now my brain hurts and I can’t stop thinking about boobies!

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is one causing the other? 🙂

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I got overwhelmed early on, why we need both kinds of people and a way to connect them.

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that seems like a good solution…

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You explained this perfectly (I did have to read this twice though). My “intuitive” thought when I read the initial numbers was somewhere under 10%, but It may be a fluke and I may have already forgotten everything. Great post!

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thanks, glad you enjoyed it. And nice job estimating the percentage – you’re a natural!

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Wow not gonna lie a lot of this went over my head LOL I’m definitely not a math person nor did I ever take stats in school but you bring up a good point about math being used in different ways in every profession (even ones you don’t think would need it)!

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I think stats is more important to take in high school than calculus, but it seems most schools don’t do it that way…

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Odds of me reading your posts? 100%

Odds of me hitting like? 100%

Did I like my stats class? A resounding NO.

Do I like or excel at math. Also a resounding NO.

What are the odds I followed the logic of this post?

You’re the stats guy. I’ll let you run the numbers.

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clever!

I’m coming up with odds close to zero 🙂

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Ok you totally lost me. Not getting through stats is the reason I don’t have a business degree. I got through everything else though. Not sure I would want a doctor reciting a bunch of statistics at me if I was worried about cancer. It would be like dealing with a robot. Oh wait my doctor is like a robot only a robot would probably remember me and why he said to come back in two weeks.

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I think losing people while trying to explain something is one of my specialties 🙂

and yes, I would want a doctor more skilled in cancer treatment than stats, but hopefully he or she is getting help with such calculations from somewhere…

soundsl like you need a new doctor…

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I tried following that calculation and your reasoning, but it all went tits up…

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seems appropriate, given the subject matter… 🙂

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Math is hard.

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but it comes in handy sometimes… 🙂

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My brain experienced statistical overload, but your discussion did make sense. I won’t need a second opinion about clicking “Like” for this post. Done!!

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thanks, Richard. It wasn’t the most exciting post I ever wrote! 🙂

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You managed to ignite some of my memories from taking a Stats class in college. And . . . I stayed awake!

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you must have had a lot of caffeine in you!

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Nah. You lost me. Sorry.

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no need to apologize; I’m used to losing people all the time when I try to explain something 🙂

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It’s the numbers that lost me. Both my children are great at maths. One did a phd in computer science. The other one did a phd in something else but lectures in stats (amongst other things) at university. They didn’t get it from me. 😂

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two PhDs! quite impressive…I’m sure they picked up a love of reading from you… 🙂

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I think they did! I didn’t really give them much choice. 😂💖

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and not having much choice turned out pretty well!

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It did! 🙂

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👍

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Well it seems I’m a stats nerd too. I calculated the rounded 8% using the second method with is intuitive [for me anyway]. Great post.

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thanks, Robbie. I like the second approach as well, since it is more intuitive. I would have picked you as a stats nerd 🙂

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I had to take statistics for a few months in high school and obviously it went great considering I am now a history major. I couldn’t quite keep up with the maths because it’s very late and also it’s maths but I do see your point.

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some people like math, some people don’t. I hope school is going well!

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Thanks and it’s not started yet but will be starting soon!

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well good luck when school does start; we’ve been in school for two weeks…

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Thanks and hope your classes are going well- it must be nice to see your students again!

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it’s nice to see them without a mask!

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Oh yeah that must be so great!

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👍

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Oh dear..at least the stats (somewhere) were correct I wasn’t the only one who thought it was a clear as mud…Maths and stats I think I flunked those in school then went on to work in a bank..funny enough I passed that entrance exam ..must have paid them enough…lol

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all that knowledge is in there somewhere! and I thought there would be some less than enthusiastic responses to this post 🙂

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If I had taken Statistics in college, I would have flunked. I now teach math in a fun way so children won’t hate it like I did.

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I thnk it is critical to instill a love of learning in children at the youngest age possible. Kudos to you for making that a priority!

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Yes! YES!! I wish you could see me subitizing, adding and subtracting using rocks, estimating with a jar full of shells…

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I’ve seen my wife prepping for such activities!

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It’s the best, isn’t it?

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I think I prefer college teaching – it’s much easier…

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😀

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Life itself is statistics

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Yes it is a good idea

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