Skill: Use probability distributions to make decisions

A probability distribution lists each possible outcome of a random variable and the probability of that outcome. To use it for a decision:

1. Check that the probabilities add to $1$.
2. Find the expected value of each alternative:

$$E(X)=\sum xP(x)$$

3. Compare the expected values using a stated criterion, such as choosing the greatest expected payoff.
4. Consider variability as a measure of risk. A larger standard deviation means outcomes are more spread out.

Example: You can choose either Plan A or Plan B for a repeated investment. Their possible payoffs are:

Plan	Payoff	Probability
A	0$	$0.20$
A	100$	$0.80$
B	60$	$0.50$
B	80$	$0.50$

Step 1: Find the expected payoff of each plan

For Plan A:

$$E(A)=(0)(0.20)+(100)(0.80)=\$80$$

For Plan B:

$$E(B)=(60)(0.50)+(80)(0.50)=\$70$$

Step 2: Use the decision criterion

If the criterion is choose the plan with the greater expected payoff, choose Plan A, because

$$\$80>\$70$$

Step 3: Recognize the risk

Plan A has more variability: it could produce either 0$or$$100$. Its standard deviation is$$40$.

Plan B’s payoffs are closer to its mean, and its standard deviation is 10$.

Therefore, Plan A has the greater expected payoff but also the greater risk. The expected payoff of 80$does not guarantee that you will receive$$80$; it represents the long-run average payoff over many similar investments. If avoiding risk is more important than maximizing expected payoff, someone might prefer Plan B instead.