Probability Risk and Decision Making

2026-01-12

Consider three complexes we can build

Payoff	Low Demand Case	High Demand Case
Small Complex (SC)	$8M, 10%	$8M, 30%
Medium Complex (MC)	$5M, 10%	$15M, 30%
Large Complex (LC)	-$11M, 5%	$22M, 10%

The percentages are called Joint Probabilities, that is, the sum of all options add up to 100%. We will need to do some work to get the probabilities.

First, we can sub across all rows to get the “probabilities” of each complex.

$$\begin{array}{rl}{P}_{SC}& =10\%+30\%=40\%\\ P_{MC}& =10\%+30\%=40\%\\ P_{LC}& =5\%+10\%=15\%\end{array}$$

Now, when calculating the expected value of each option, we need to use Conditional Probabilities. For example, for the Small Complex (SC), we have:

$$\begin{array}{rl}{P}_{LD/SC}& =\frac{10\%}{40\%}=25\%\\ P_{HD/SC}& =\frac{30\%}{40\%}=75\%\\ E{V}_{SC}& =(8M\ast 25\%)+(8M\ast 75\%)=8M\end{array}$$

The last two can be calculated similarly:

$$\begin{array}{rl}E{V}_{MC}& =11.67M\\ E{V}_{LC}& =11M\end{array}$$

Possible Decisions

There are three methods that we can base our decisions on.

1. Seek the highest expected value

In this case, we would pick to build the medium complex.

2. Seek the best, worst-case solution

In this case, we would pick the build the small complex as that

3. We apply HARA utility functions to determine the best option.