(15%) Q5) A geotechnical engineer is evaluating the performance of a drainage system composed of
10 independent soil columns, each with a hydraulic conductivity K (in m/s) that follows a log-normal
distribution. The mean hydraulic conductivity is $\mu_K = 1.5 \times 10^{-4}$ m/s, and the coefficient of variation is
0.6. The system is considered to function properly if at least 7 out of the 10 columns have a conductivity
greater than $1 \times 10^{-4}$ m/s.
(4%) a) Calculate the probability that a single soil column has a conductivity greater than $1 \times 10^{-4}$ m/s.
(3%) b) Assuming independence, model the number of functioning columns (i.e., those with $K > 1 \times$
$10^{-4}$) as a binomial random variable. What is the probability that at least 7 out of 10 columns function
properly?
(2%) c) Find the 95th percentile of the hydraulic conductivity.
(3%) d) If the engineer wants to ensure that the system functions with at least 95% probability, what
should the minimum probability of a single column functioning be?
(3%) e) Based on your answer to (d), what should the mean hydraulic conductivity be (assuming the
same COV) to meet this system reliability requirement?
