## 2.3 Find the expected value (EV)

You are now ready to evaluate the relative merits of each decision alternative. Expected value (EV) is the way to combine payoffs and probabilities for each node. The higher the EV, the better a particular decision alternative on average when compared to the other alternatives in the decision tree.

The method for calculating EV differs slightly based on the type of node. In figure 2.3 we consider chance nodes first.

You calculate the EV for any chance node by summing together all the EVs for each branch that is connected to the node. The general formula for calculating EV at any chance node is:
EV chance node = EV branch1 + EV branch2 + . . . + EV branchN

• In the Really Big Ideas scenario, if the smoke and fire detector is successful, the EV is the payoff (profit) multiplied by its probability, or \$900,000 x 0.5 = \$450,000.

• The EV if the fire detector project fails is (-\$100,000) x 0.5 = (-\$50,000).

• The EV for the decision to develop the smoke and fire detector (incorporating both success and failure) is the sum of the EV for all the eventualities.
EVchance node = (EV success + EV failure) = \$450,000 + (-\$50,000) = \$400,000.

• Similarly, the EV for the decision to develop the motion detector is given by
EV = (\$390,000 x 0.8) + [(-\$10,000) x 0.2] = \$310,000.

Write the EV for each node near that node, as shown in figure 2.3.

Fig 2.3 Expected value (EV) is the sum of all the combined payoffs and probabilities for each node.

The smoke and fire detector project has a higher EV than the motion detector. You can report the analysis with these summarized presentation points:
• The smoke and fire detector is the better project to develop, despite the greater risk. The significantly larger anticipated profits make the risk more acceptable than the competing project.
• The motion detector is less risky, but also significantly less profitable. With the given profit expectations the motion detector project does not overcome the expected value of its rival project.

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