Question

if a company considers 10 different future states of nature to be equally likely, and each of the 10 has a different value, how can that company calculate the expected value

Answer 1

The company can calculate the expected value by taking the weighted average of the possible future states of nature. Each future state's probability is multiplied by its corresponding value, and the results are summed up.

Step-by-step explanation:

To calculate the expected value when there are 10 equally likely future states of nature with different values, we use the formula:

`\[E(X) = \sum_(i=1)^(10) P(X_i) \cdot V(X_i)\]`

where E(X) is the expected value,
`\(P(X_i)\)` is the probability of the ith future state of nature, and
`\(V(X_i)\)` is the corresponding value. Since each of the 10 states is equally likely, the probability
`(\(P(X_i)\))` for each state is
`\(1/10\)`.

Let's assume the values for the 10 states are
`\(V(X_1), V(X_2), ..., V(X_(10))\)`. The expected value calculation becomes:

`\[E(X) = (1)/(10) \cdot V(X_1) + (1)/(10) \cdot V(X_2) + ... + (1)/(10) \cdot V(X_(10))\]`

This simplifies to:

`\[E(X) = (1)/(10) \cdot (V(X_1) + V(X_2) + ... + V(X_(10)))\]`

So, the expected value is the average of the 10 different values, each multiplied by its respective probability. This calculation provides a useful metric for decision-making, representing the average outcome considering all possible future states of nature.