Logical fallacy: Conjunction fallacy

It is a theorem in logic that \displaystyle P(A) \geq P(A \mbox{ AND } B), that is the probability of some event A occurring is always higher than the probability of both A and B occurring, unless B always occurs when A occurs in which case they are equal. In other words, the chances of something happening goes down as you add detail, even though the addition of detail may make a story more convincing.

What is the probability you’d assign to the statement “A Democrat will win the US Presidential Election in 2016”? For argument’s sake, let’s say you decide it’s a coin flip, 50-50. Now, what is the probability that Hillary Clinton will win the 2016 election? Since she herself is a Democrat, the upper bound of such a prediction given the previous prediction must be 50%, since you can’t have the probability of Hillary Clinton and a Democrat winning be less than a Democrat winning (since some other Democrat might win). The probability goes down further if you say something like “Hillary Clinton, a Democrat, will win the 2016 Presidential Election while carrying Ohio but losing in North Carolina.” As detail goes up, the probability must go down even if the story sounds more convincing with more detail.

When Intel fabricates a silicon wafer, a bunch of chips are patterned onto the disk and are then cut out. Not all chips work in the end, so some fraction pass quality and control and are sold to customers, the yield. Say you were in charge of designing the process for this wafer, and it turned out that 80 steps were required (clean 1, deposit 1, photo 1, etch 1, … clean 2, deposit 2, etc). You go to the lead process engineer and ask what the increase in failure rate for each step is, and he or she assures you that each step is very high quality, 99% chance of no problems occurring. No problem right? 99% is a probability we can be pretty confident in. But when each step (we’re ANDing the steps together) has this probability applied to it, by the end you’ve lost over half your chips since \displaystyle 0.99^{80} \approx 0.48.

Even if you are very confident in each detail, chaining many of them together necessarily lowers the total probability with each new addition. When making predictions or evaluating predictions about the future, keep the details to a minimum as those explanations strictly dominate detail-heavy stories.

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