The following quote is from Paul J. Nahin\'s Duelling Idiots and Other Probabili

Question

The following quote is from Paul J. Nahin's Duelling Idiots and Other Probability Puzzlers. A far more interesting application ... was reported in the Boston Globe 0n January 24, 1992 (p. 3), as part of the retrospective analyses then being conducted on the Gulf War. In particular, the Pentagon had gone on record with a claim that the Patriot antiaircraft missile system had "successfully engaged over 80 percent" of the Scud missiles Iraq had launched at Saudi Arabia. An MIT physicist, Theodore Postol, disputed that claim; it was a remarkable claim, too, as the Patriot was designed to counter relatively slow manned aircraft, not supersonic ballistic missiles. Postol based his skepticism on what he saw after watching videotapes of fourteen engagements. There were thirteen misses and one probable hit. The Globe article ended with this quote from Professor Postol: "What are the odds 1 would see 13 misses and one hit if the Patriot was successfully shooting down 80 percent of the Scuds?" Answer Postol's question.

Explanation / Answer

Let X = Number of hits made by the missile system

Note all the attempts of hit are independent.

If we assume Pentagons statement to be true then the probability of a hit is 0.8.

Thus X follows binomial distribution with probability of success = 0.8

Now for Postols observation; the number of engagements were 14.

Thus X follows B(n=14, p=0.8)

Postols Statement : What are the odds I would see 13 misses and one hit if Patriot was susseccfully shotting down 80%of the Scuds

This can be written as a Probabilty statement as follows,

P[X=1] ; where X follows B(n=14,p=0.8)

We know that, P[X= x ] = nCx * (p)x * (q)n-x

Thus, P[X= 1] = 14C1 * (0.8) * (0.2)13

  = 9.17504 x 10-9  

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