10.2 Probability of Failure in a Portfolio
147
Table 10.1 Probability of failure of theoretical sample portfolios of identical dams
Tailings dams World wide
Poor dams
Hydro dams world wide
Very poor dams
Best dams world-wide
Pf dam, identical for the three dams
T
10
100
1000
10000
100000
1000000
P=1/T
single
dam
0.1
0.01
0.001
1.00E-4
1.00E-5
1.00E-6
Ptot(3)
0.271
0.0297
0.003
3.00E-4
3.00E-5
3.00E-6
Ptot(10)
0.651
0.0956
0.00996
1.00E-3
1.00E-4
1.00E-5
Ptot(100)
1
0.634
0.0952
9.95E-3
1.00E-3
1.00E-4
of failure of the selected type of dam. The coloured domains in Table 10.1 are based on
statistics related to the world-wide dams portfolio we developed in (Oboni and Oboni
2013). For example, poor dams (orange colour) have p f between, say, 1/10 = 0.1 and
1/100 = 0.01 = 1.0
−2 . Excellent dams (best dams world-wide, blue colour) have p f
between, say, 1/100,000 = 1.0
−5 and 1/1,000,000 = 1.0
−6 . The other “categories”
lie in-between as indicated. Ptot (3), Ptot (10), and Ptot (100) indicate the p fS for
portfolios of 3, 10, 100 dams of identical p f = 1/T single dam.
The same information can be show in graphic form (logarithmic scales) in
Fig. 10.5. Obviously, if the quality of the dams is poor to very poor the chance
of a system failure is high, even for a small portfolio (3 dams).
However, as soon as the quality of the dams reaches world-wide standards for
hydro-dams (not agricultural dams in the United States, for example, or decrepit
structures anywhere in the world) the system’s probability remains below 1/100 =
0.01 = 1.0
−2 and for portfolios of, for example, 10 dams or fewer, it is below 1/1000
= 0.001 = 1.0
−3 .
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
0.000001
0.00001
0.0001
0.001
0.01
0.1
Probability of System of IdenƟcal
Elements
Probability of Failure of a Sigle Dam
Ptot(3)
Ptot(10)
Ptot(100)
Fig. 10.5 Probability of failure of theoretical sample portfolios of identical dams