Of chains and funnels

What to make of chains and funnels? And, if I also stick in anchors, does it help?

What I'm actually talking about is conjunctive events, disjunctive events, and anchor bias:

The general tendency to overestimate the probability of conjunctive events leads to unwarranted optimism in the evaluation of the likelihood that a plan will succeed or that a project will be completed on time. Conversely, disjunctive structures are typically encountered in the evaluation of risks. A complex system, such as a nuclear reactor or a human body, will malfunction if any of its essential components fails.

Daniel Kahneman and Amos Tversky

Judgment Under Uncertainty: Heuristics and Biases

• Conjunctive events are chains of event for which every link in the chain must be a success or the chain fails. Success of the chain is the product of each link's success metric. In other words, the chain's success probability degrades geometrically (example: chain of 'n' links, each with probability 'p', has an overall probability of p*p*p* .... for 'n' p's.)
    
• Disjunctive events are independent events, all more or less in parallel, somewhat like falling in a funnel, such that if one falls through (i.e, failure) and it's part of a system, then the system may fail as a whole. In other words, if A or B or C goes wrong, then the project goes wrong.

Fair enough. Where does the anchor come in?

Anchoring refers to the bias introduced into our thinking or perception by suggesting a starting value (the anchor) but then not adjusting far enough from the anchor for our estimate to be correct. Now in the sales and marketing game, we see this all the time. Marketing sets an anchor, looking for a deal in the business case; the sales guy sets an anchor, hoping not to have to give too much away post-project. The sponsor sets an anchor top down on the project balance sheet, hoping the project manager will accept the risk; and the customer sets anchors of expectations.

But in project planning, here's the anchor bias:

• The likely success of a conjunctive chain is always less than the success of any link
• The likely failure of a disjunctive funnel is always greater than the failure of any element.

Conjunctive chains are products of numbers less than 1.0.

• How many of us  would look at a 7 link chain of 90% successes in each link and realize that there's less than one 1 chance in 2 that the chain will be successful? (probability = 0.48)

Disjunctive funnels are more complex. They are the combinations (union) of independent outcomes net of any conjunctive overlaps (All combinations of OR less all AND). In general the rules of combinations and factorials apply.

• How many of us would look at a funnel of 7 objects, each with likely 90% success (10% failure) and realize that there's better than 1 chance in 3 that there will be 1 failure among 7 objects in the funnel? (probability = 0.37 of exactly 1 failure)*

 The fact is, in the conjunctive case we fail to adjust downward enough from 90%; in the disjunctive case we fail to adjust upward from the 10% case.  Is it any wonder that project estimates go awry?

*This is a binomial combination of selecting exactly 1 from 7, where there are 6 conjunctive successes and 1 conjunctive failures: factorial (7 take 1) *conjunctive failure * conjunctive success

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