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Conditional probability is hard -- but teaching it *shouldn't* be!

So, consider these two problems: 

A. Which is more difficult?

B. Which is it easier to teach someone to do correctly?

My answers: BAYES is more difficult but also easier to each someone to do correctly. 

Does that seem plausible to you? I won't be surprised if you say no, particularly if your answer reflects experience in seeing how poorly people do with conditional probability problems.

But if you disagree with me, I do want to challenge your sense of what the problem is

Okay, so here are some data.

For sure, BAYES is harder.  In a diverse sample of 1,000 adults (over half of whom had either a four-year college or post-graduate degree), only 3% got the correct answer (50%). For COVARY, 55% percent got the correct answer (“patients administered the new treatment were not more likely to survive”).

This is not surprising. BAYES involves conditional probability, a concept that most people find very counterintuitive.  There is a strong tendency to treat the accuracy rate of the witness’s color discernment-- 90% --  as the likelihood that the bus is blue.  

That was the modal answer—one supplied by 34% of the respondents—within the sample here. This response ignores information about the base rate of blue versus green buses.  

Another 23% picked 10%--the base rate frequency of blue buses. They thus ignored the additional information associated with the witness’s perception of the color of the bus.

How to combine the base rate information with the accuracy of the witness’s perception of color (or their equivalent in other problems that involve the same general type of reasoning task) is reflected in Bayes’s Theorem, a set of logical operations that most people find utterly baffling.

COVARY is a standard “covariance detection” problem.  It’s not as hard as BAYES, but it’s still pretty difficult!

Many people (usually most; this fairly well educated sample did better than a representative sample would) use one of two heuristics to analyze a problem that has the formal characteristics of this one (Arkes & Harkness 1983).  The first, and most common, simply involves comparing the number of “survivors” to the number of “nonsurvivors” in the treatment condition.  The second involves comparing in addition the number of survivors in the treatment and the number of survivors in the control.

Both of these approaches generate the wrong answer—that patients given the new treatment were more likely to survive than those who didn’t receive it—for the data generated in this hypothetical experiment.

What’s important is the ratio of survivors to nonsurvivors in the two experimental groups.  In the group whose members received the treatment, patients were about three times more likely to survive (223:75 = 2.97:1).  In the untreated group, however, parents were just over five times more likely to survive (107:21 = 5.10:1).

Pretty much anyone who got the wrong answer can see why the correct one is right once the difference in the “likelihood ratios” (which is actually an important common element in conditional probability and covariance problems) is pointed out. 

The math is pretty tame (a fifth grader should be able to handle it), and the inferential logic (the essence of the sort of causal inference strategy that informs controlled experimentation) pretty much explains itself.

The reason such a significant number of people get the answer wrong is that they don’t reliably recognize that they have to compare the ratios of positive to negative outcomes. They effectively succumb to the temptation to settle for “hypothesis-confirming” evidence without probing for the disconfirming evidence that one can extract only by making use of all the available information in the 2x2 contingency table.

Now, why do I feel that it is nevertheless easier to teach people how to solve conditional probability problems of the sort reflected in BAYES than to teach them how to reliably solve covariance-detection ones of the sort reflected in COVARY?

The answer has to do with what someone has to learn to consistently get the problems right.

Doing conditional probability problems is actually easy once one grasps why the base rate matters—and enabling someone to grasp that turns out to be super easy too with the right pedagogical techniques.

The most important of these is to illustrate how a conditional probability problem can be conceived of as a population-sampling one (Spiegelhalter, Pearson & Short 2011).

In BAYES, we are told that 90% of the buses that could have struck Bill are green, and 10% of them are blue.

Accordingly, if we imagine a simulation in which Bill was hit by 100 city buses drawn at random, we’d expect him to be run down by a green bus 90 times and a blue one 10 times.

If we add Wally to the simulation, we’ll expect him correctly to perceive 81 or 90% of the 90 green buses that struck Bill  to be green and incorrectly perceive 9 (10%) of them to be blue.

Likewise, we’ll expect him to correctly perceive 9 of the 10 blue buses (90%) that hit Bill to be blue, but incorrectly perceive 1 of them (10%) to be green.

Overall, then, in 100 trials, Wally will perceive Bill to have been hit 18 times by a blue bus. Nine of those will be cases in which Wally correctly perceived a blue bus to be blue.  But nine will be cases in which Wally incorrectly perceived as blue a bus that was in fact green.

Because in our 100-trial simulation, the number of times Wally was correct when he identified the bus that hit Bill as blue is exactly equal to the number of times he was incorrect, Bill will have been hit by a blue bus 50% of the time and by a green one 50% of time in all the cases in which Wally perceives Bill was hit by a blue bus.

This “natural frequency” strategy for analyzing conditional probability problems has been shown to be an effective pedagogical tool in experimental studies (Sedlmeier & Gigerenzer 2001; Kurzenhäuser & Hoffrage 2002; Wheaton & Deshmuk 2009). 

After using it to help someone grasp the conceptual logic of conditional probability, one can also connect the steps involved to a very straightforward rendering of Bayes’s Theorem: prior odds x likelihood ratio = revised (posterior) odds.

In this rendering, the base rate is represented in terms of the odds that a particular proposition or hypothesis is true: here, independently of Wally’s observation, we’d compute the odds that the bus that struck Bill was green at 10:90 (“10 in 100”) or 1:9.

The new information or evidence is represented as a likelihood ratio, which reflects how much more consistent that evidence is with the hypothesis or proposition in question being true than with its negation (or some alternative hypothesis) being true.

Wally is able correctly to distinguish blue from green 90% of the time.

So if the bus that struck Bill was in fact blue, we’d expect Wall to perceive it as blue 9 times out of 10, whereas if the bus that struck Bill was in fact green, we’d expect Wally to perceive it as blue only 1 time out of 10. 

Because Wally is nine times (9 vs. 1 or 90% vs. 10%) more likely to perceive a bus was “blue” when it was truly blue than when it was in fact green, the likelihood ratio is 9.

“Multiplying” the prior odds by the likelihood ratio involves computing the product of (1) the element of the odds expression that corresponds to the hypothesis  and (2) the likelihood ratio value. 

Here the prior odds were 1:9 that the bus that struck Bill was blue.  Nine (likelihood ratio) times one (from 1:9) equals 9

The revised odds that the bus that struck bill was blue is thus 1:9 x 9 = 9:9 or 1:1, which is equivalent to 50%.

I’m not saying that one exposure to this sort of exercise will be sufficient to reliably program someone to do conditional probability problems.

But I am saying that students of even middling levels of numeracy can be expected over the course of a reasonable number of repetitions to develop a reliable facility with conditional probability. The “natural frequencies” representation of the elements of the problem makes sense, and students can see which parts of that conceptualization map onto the “prior odds x likelihood ratio = revised odds” rendering of Bayes’s theorem and why.

If you want to make it even easier for this sort of lesson to take hold, & related hardwiring to settle in, give your students this cool Bayes's calculator.

Students can’t be expected, in contrast, to see why any of the other more complex but logically equivalent rendering of Bayes’s Theorem actually makes sense.  They thus can't be expected to retain them, to become adept at heuristically deploying them, or to experience the sort of improvement in discernment and reasoning that occurs as one assimilates statistical concepts.  

Teachers who try to get students to learn to apply these formalisms, then, are doing a shitty job!

Now what about covariance?

Actually, there’s really nothing to it from an instructional point of view.  It explains itself, as I said.

But that’s exactly the problem: facility with it is not a matter of learning how to do any particular thing.

Rather it is a matter of reliably recognizing when one is dealing with a problem in which the sort of steps necessary to detect covariance have to be done.

The typical reaction of someone when it's pointed out that he or she got the covariance problem wrong is an instant recognition of the mistake, and the sense that the error was a result of an uncharacteristic lapse or even a “trick” on the part of the examiner. 

But in fact, in order to make reliable causal inferences based on observation in their everyday life, people will constantly be required to detect covariance.  If they are unable to see the need for, or just lack the motivation to perform, the necessary operations even when all the essential information has been pre-packaged for them into a 2x2 contingency table, then the likelihood that they will lapse into the defective heuristic alternative when they encounter covariance-detection problems in the wild is very very high (Stanovich 2009).

How likely someone is to get the right answer in the covariance problem is associated with their numeracy. The standard numeracy scale (e.g., Peters et al. 2006) is a measure not so much of math skill as of the capacity to reliable recognize when a quantitative reasoning problem requires one or another type of effortful analysis akin to what's involved in detecting covariance.

Frankly, I’m pessimistic that I can instill that sort of capacity in students.  That's not because I have a modest sense of my abilities as a teacher.  It’s because I have due respect for the difficulty that many indisputably great researchers and teachers have encountered in trying to come up with pedagogical techniques that are as successful in imparting critical reasoning dispositions in students as the “natural frequencies” strategy is for imparting a reliable facility in them to do conditional probability problems.

Of course, in order for students to successfully use the “natural frequencies” strategy and—after they become comfortable with it—the prior odds x likelihood ratio = revised odds rendering of Bayes theorem, they must reliably recognize conditional probability problems when they see them. 

But in my experience, at least, that’s not a big deal. When a conditional probability problem makes its appearance, one is about as likely to overlook it as one is to fail to notice that a mother black bear w/ its cub or a snarling honey badger has appeared along side the trail during a hike in the woods.

Which then leads me to the question, how can it be that only 3% of a sample as well educated and intelligent  as the one I tested can get do a conditional probability problem as simple as the one I put in this battery?

Doesn't that mean that too many math teachers are failing to use the empirical knowledge that has been developed by great education researchers & teachers?

Or am I (once again; it happens!) missing something?


Arkes, H.R. & Harkness, A.R. Estimates of Contingency Between Two Dichotomous Variables. J. Experiminal Psychol. 112, 117-135 (1983).

Kurzenhäuser, S. & Hoffrage, U. Teaching Bayesian reasoning: an evaluation of a classroom tutorial for medical students. Medical Teacher 24, 516-521 (2002).

Peters, E., Västfjäll, D., Slovic, P., Mertz, C.K., Mazzocco, K. & Dickert, S. Numeracy and Decision Making. Psychol Sci 17, 407-413 (2006).

Sedlmeier, P. & Gigerenzer, G. Teaching Bayesian reasoning in less than two hours. Journal of Experimental Psychology: General 130, 380-400 (2001).

Spiegelhalter, D., Pearson, M. & Short, I. Visualizing Uncertainty About the Future. Science 333, 1393-1400 (2011).

Stanovich, K.E. What intelligence tests miss : the psychology of rational thought (Yale University Press, New Haven, 2009).

Wheaton, K.J., Lee, J. & Deshmukh, H. Teaching Bayesian Statistics To Intelligence Analysts: Lessons Learned. J. Strategic Sec. 2, 39-58 (2009).



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Reader Comments (4)

However, Willy is the product of more than 4 billion years of evolution.
Willy's information processing and his senses are under hormonal control, so that when stressed, there are huge changes in his perception; generally part of what is known as the flight or fight reflex.
When Willy observed the accident blood-flow to his retina, optic nerve and visual cortex increased massively upregulating information delivery and processing. To Willy, it appeared that time itself had slowed, and that he was watching all things happening in slow motion.
Such responses are a species motif and demonstrate the usefulness of responsive information acquisition and analysis systems for identifying camouflaged predators. When tested under normal conditions, Willy did miss identify colors of busses some 10% of the time, however, when tested in a heightened alert state, this fell to <0.5%.
Anyone using Baysian analysis in such as case would be mistaken.

August 21, 2014 | Unregistered CommenterDocMartyn


I think w/ education, humans can teach themselves to reason better than they did, say, 100,000 yrs ago (has our basic cognitive apparatus evolved since then? I'm not sure! But doesn't matter: w/ education, we can improve on what we are endowed w/ no matter how recent the latest system update).

Indeed, a *huge* amount of education consists in inculcating habits of mind that resist tendencies to err that no doubt were themselves "adaptive."

*That* ability -- to use reason to train our minds -- is part of the system we have acquired through evolution, too.

So let's make the best of it!

August 22, 2014 | Registered CommenterDan Kahan

There is a typo - in the following sentence, I assume the 90% should be 10%

Another 23% picked 90%--the base rate frequency of blue buses. They thus ignored the additional information associated with the witness’s perception of the color of the bus.

August 23, 2014 | Unregistered CommenterAllan

Yes-- thx! Teachers in our nation have failed to teach good proofreading skills. Outrageous!

August 23, 2014 | Registered CommenterDan Kahan

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