Second Eggs

I read Tim Harford's book Undercover Economist; in the second edition is an argument about
mortgage-backed securities and further derivatives.

The author must have included this on the notion that some financial instruments which were involved in the world's recent financial crisis are similar to "second eggs."  He mentions that mortgage-backed securities were derivative on mortgages being generally safe, but that further derivatives were taken as well.  And so he suggests we consider a toy model in which you are offered an investment which we'll call an egg.  The investment will probably deliver value.  On the other hand, it might become worthless.  We estimate that eggs go bad with probability p.  From eggs, we get an investment called "second eggs" which names a "carton" of 6 eggs and pays off if the number of bad eggs turns out to be 0 or 1.  From e second eggs, we can of course get a carton of second second eggs, and so on.  

This emphasizes that while p represents risk, estimating p wrong is also a risk.  And yet, a basic notion in probability theory is that we have to guess something and then reason to a conclusion.  We have to guess p, and then see what the consequences are.  Or we can guess a distribution for p and see what the consequences are.  But what are the consequences of guessing the wrong distribution?  Probability theory should hope that the recursion stops here... that a distribution of distributions for p is not any different from a distribution for p.  And we could hope that there are only two sorts of error:

-- the probability p that an investment goes sour;

-- the error in estimating p; call it p'.   

He points out that if eggs go bad with probability p, then "the second egg to go bad" is an investment which goes sour with probability q.  He then points out two interesting things: if p is below 5%, then q < p.  I.e., the second bad egg in a batch of six is a safer bet than buying a single egg.  On the other hand, if p is 10% or more, then q > p.  Supposing that we no extra information about second eggs than that they are ... second eggs... then to estimate q we must minimize p'.  If we can't guarantee that p < 5%, then we can't guarantee that q < p.  I made this table just to check.  In the left column, the probability that one egg is bad is 5%; in the right column it is 10%.  Then we calculate the probability that the number of eggs which are bad is > 2 in a batch of 6. 

Example from Tim Harford book Undercover Economist, version 2.
batch six eggs, and rank them with the worst ones first (perhaps this is revealed later).
probability that one egg is bad:				0.05	0.1
distribution of the number of bad eggs:
			0	0.735091891	0.531441
			1	0.232134281	0.354294
			2	0.030543984	0.098415
			3	0.002143438	0.01458
			4	8.46094E-05	0.001215
			5	1.78125E-06	0.000054
probability that the first egg is bad:				0.264908109	0.468559
probability that the second egg is bad:				0.032773828	0.114265
probability that the third egg is bad:				0.002229844	0.01585
probability that the fourth egg is bad:				8.64063E-05	0.00127
probability that the fifth egg is bad:				1.79688E-06	5.5E-05
probability that the sixth egg is bad:				1.5625E-08	1E-06
(The author mistakenly rounds 0.26490810 to 0.27.)
batch six SECOND eggs, and rank them with the worst ones first (perhaps this is revealed later).
probability that one second egg is bad:				0.032773828	0.114265
distribution of the number of bad second eggs:
			0	0.818781906	0.482862
			1	0.166463346	0.373752
			2	0.014101255	0.12054
			3	0.000637082	0.020734
			4	1.61903E-05	0.002006
			5	2.1944E-07	0.000104
probability that the first egg is bad:				0.181218094	0.517138
probability that the second egg is bad:				0.014754748	0.143386
probability that the third egg is bad:				0.000653493	0.022846
probability that the fourth egg is bad:				1.6411E-05	0.002112
probability that the fifth egg is bad:				2.20679E-07	0.000106
probability that the sixth egg is bad:				1.23926E-09	2.23E-06
batch six second second eggs, and rank them with the worst ones first.
probability that one second egg is bad:				0.014754748	0.143386
distribution of the number of bad second second eggs:
			0	0.914673513	0.395103
			1	0.082187318	0.396811
			2	0.003077034	0.166053
			3	6.1441E-05	0.03706
			4	6.90092E-07	0.004653
			5	4.13385E-09	0.000312
probability that the first egg is bad:				0.085326487	0.604897
probability that the second egg is bad:				0.003139169	0.208086
probability that the third egg is bad:				6.21353E-05	0.042033
probability that the fourth egg is bad:				6.94236E-07	0.004973
probability that the fifth egg is bad:				4.14417E-09	0.00032
probability that the sixth egg is bad:				1.03176E-11	8.69E-06