# John Brown University Probability Questions

As your first engineering job, you are tasked to assess the failure probability for a newbridge connecting the mainland with a remote island near Tierra del Fuego (Land of Fire).
Given the relative proximity of the structure to Los Andes, it is expected to be affected by
multiple earthquakes (E) during its lifespan. Also, given that the main purpose of the bridge
is to allow trucks to supply the remote island, it will also be affected by heavy vertical
loads (V) as the trucks tend to travel in tandem to support each other in case of emergency.
Upon further consulting with geologists and earthquake engineers, the earthquakes in the
zone can be simplified as belonging to one of two categories: severe earthquakes (E) and
mild earthquakes (EM). The vertical load, instead, can be classified into three classes: high
(VH), medium (VM), and low (V₁). Finally, the bridge main supports can fail by any of the
following two failure mechanisms: shear (S) and flexure (F).
The conditional probabilities of the different failure mechanisms given a hazardous event
are given below:
• P(SES) = 0.3
• P(SIEM) = 0.03
P(F|VH) = 0.1
P(F|VM) = 0.02
P(F|VL) = 0.005
P(S|VH) = P(SVM) = P(S|V₂) = P(F|Es) = P(F|EM) = 0Additionally, the probability of occurrence for a strong or mild earthquake is 0.10 and
0.90 respectively. The probability of occurrence for a high, medium, or low vertical load
due to the trucks is 0.08, 0.22 and 0.7 respectively. Please answer:
a. What does the expression
P(S|VH) = P(SVM) = P(S|V₂) = P(F|Es) = P(FEM) = 0
mean in practice? For the answer, relate the failure mechanism to the different
hazardous event.
b. Determine the probability that an earthquake causes a shear failure.
c. Determine the probability that the traffic of the trucks causes a flexure failure?
d. Let’s suppose that a flexure failure occurs, what is the probability that the traffic load
was medium?
e. Now, let’s suppose that a shear failure occurs, what is the probability that the
earthquake was strong?

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