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Be aware that:
  • The assumption of independence is a strong assumption.
  • Independence of events must be verified.
  • You must identify and account for lurking variables.

1.  The assumption of independence is a strong assumption.

Interactivity

Investigate independence using the interactive Venn Diagram Applet

which reports conditional probability. 

  • Drag A until P(A|B) = P(A). = 0.075. 
  • Note that P(B|A) = P(B) = 0.2 as well. 
  • For all other values of P(A and B) the events are dependent.

2. Independence of events must be verified.

Example

In January 1992, the Challenger space shuttle exploded a minute after takeoff.  The O-rings meant to contain burning gases during liftoff failed. Engineers had assumed the O-rings would function independently with failure rate of 0.023.
  • With independence: P(both fail) = (.023)(.023) = (.000529). 
  • So P(at least one works) = 1-.000529 = .999471 for each joint with an O-ring.
  • However there was a common failure mode; the cold and vibrations caused both O-rings to fail: P(2nd O-ring failed given first O-ring failed) = 1 not .023

When the shuttle was redesigned after the accident, a 3rd O-ring was added which is truly independent. So the P(failure) is now very small, but not zero.

3.   You must identify and account for lurking variables.
When data are aggregated over a lurking variable the results may reverse.  This is known as Simpson's Paradox.

[Stan: please install the linking arrow to 'Example 1', that David Schaller has provided, here -  pointing right]

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