### Overestimating Conjunctive Probabilities and Underestimating Disjunctive Probabilities

Here's an interesting finding from cognitive bias researchers (like Tversky and Kahneman): people tend to overestimate conjunctive probabilities and underestimate disjunctive probabilities.

This theory could explain why investors are (over)confident that a company they've bet on will succeed. As Kahneman explains in Thinking, Fast and Slow:

I'll bet that when people think of a plan that only needs seven things to succeed, each with a 90 percent chance of success, they essentially mentally average the probabilities to come up with a roughly 90 percent conjunctive probability. When at that point, it's actually more likely than not (0.9^7=0.48) that the plan will not succeed!"Biases in the evaluation of compound events are particularly significant in the context of planning. The successful completion of an undertaking, such as the development of a new product, typically has a conjunctive character: for the undertaking to succeed, each of a series of events must occur. Even when each of these events is very likely, the overall probability of success can be quite low if the number of events is large. The general tendency to overestimate the probability of conjunctive events leads to unwarranted optimism..."

So, looking at an investment like Molycorp, the backers need to be able to make a conjunctive probability assessment. The company needs to make many systems work along a chain from ore to refined product. Some of those systems, like the hydrochloric acid plant and the leach tanks, have had what are now longstanding, intractable problems. My skepticism last November about Molycorp's ability to produce (see 1,2,3), not to mention the price that rare earth metals would bring, seems in retrospect like a good conjunctive probability assessment.

In contrast, people underestimate disjunctive probabilities, as Kahneman explains:

It seems like judging failure is almost always going to be disjunctive (many possible causes) and success almost always conjunctive. Add in people's existing biases for optimism, and you can see why investors are mostly bulls and bears are so rare."A complex system, such as a nuclear reactor or a human body, will malfunction if any of its essential components fails. Even when the likelihood of failure in each component is slight, the probability of an overall failure can be high if many components are involved."

## 2 comments:

I agree when the events are independent, but there is also research out there that shows people under-estimate the inter-dependence of events, esp rare events. I think that Mr. Taleb would agree.

So we actually see two seemingly contradictory effects: we overestimate events of conjunctive probabilities that are independent (and not too rare), but we under-estimate the conjunctive probabilities of rare events by assuming independence.

Great post!

This reminds me of something I thought of heading into the housing bubble bust.

Picture an unemployment rate rising to 10% in a world where many households live paycheck to paycheck.

In a one earmer household, there's roughly a 10% chance of having a serious problem making the mortgage payment.

In a two earner household, if either person loses a job, there is a serious problem. Those odds are 19%, all else being equal.

Relying on two people to both have a job in order to make the mortgage payment increases the risk off a serious problem substantially, and yet that is exactly what is/was encouraged.

And people wonder what went wrong?

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