Two small rivers A and B meet and become a larger river C. For a new housing pro

Question

Two small rivers A and B meet and become a larger river C. For a new housing project, the flood level, defined as the water level above the mean level of river C, is under consideration. Past records indicate that a 1-ft change in the water level of river A causes a 0.5-ft change in the water level of river C, and a 1-ft change in the water level of river B causes a 0.25-ft change in the water level of river C. The flood levels of rivers A and B can be estimated precisely. However, they are equally likely to be 0, 1, 2, and 3 ft for both rivers. Assuming statistical independence of flood levels between rivers A and B, what is the probability that the flood level of river C will exceed 1.75 Ft?

Explanation / Answer

The change in level of C due to A with the required probabilities is given as:

The change in level of C due to B with the required probabilities is given as:

Now the combined effect of A and B on C can be modelled as:

All the 16 possibilities in the above grid are equally likely and have a probability of 0.25*0.25 = 0.0625

Therefore as the number of grid points where the C level exceed 1.75 ft is 2 points, therefore the required probability here is 0.0625*2 = 0.125

Therefore 0.125 is the required probability here.

A=a 0 1 2 3 P(A=a) 0.25 0.25 0.25 0.25 C level 0 0.5 1 1.5

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