Question

A landfill containing system is used to prevent leakage of contaminants from the landfill into the soil
stratum after the rainfall. The system consists of a thick layer of clay (highly impermeable material)
placed between the landfill and the surrounding soil stratum, and a layer of synthetic material called
geomembrane placed above the clay material to provide additional protection.
The quality of workmanship during construction may not be completely satisfactory. First, the clay
might have been compacted poorly; second, the geomembrane might have holes puncture by sharp
stones that were not detected during inspection. Additionally, extremely heavy rainfall could happen
during the operation of the landfill, which could induce excessive pore pressure on the geomembrane
and clay layers.
The site engineer believes that leakage will happen “during extremely heavy rainfall” when the clay was
not well compacted or there were holes in the geomembrane (call this event E1). Leakage can also occur
“during ordinary rainfall”, but only when clay was not well-compacted, and the geomembrane
contained holes (call this event E2).
The site engineer has determined the following:

• The probability of the clay not being well compacted (C) is 15% ( i.e P(C) = 0.15 )

• The probability of the geomembrane containing holes (H) is 25% ( i.e P(H) = 0.25 )

• The probability of having extremely heavy rainfall (R) is 15% ( i.e P(R) = 0.15 )

The quality of the construction has no effect on the rainfall, but if the geomembrane contained holes,
the probability of not having well compacted clay increases to 50%.

a. Express E1 and E2 in terms of the symbols defined above. [2 marks]

b. Determine the probability of events E1 and E2. [4 marks]

c. Are C and H mutually exclusive? Are C and H statistically independent? Are events E1 and E2
mutually exclusive? Provide explanations to support your answers. [2 marks]

d. Determine the probability of leakage for this landfill containment system. [2 marks]

Answer 1