20.1: Uncertainty and Expected Value

People face uncertain situations. Uncertainty arises in situations where future outcomes are unknown and influenced by chance or external factors. A college student may get a high-paying job as soon as they graduate in the future or remain unemployed for a long time.

Another example of uncertainty is a college basketball team playing the final game of a championship. The team may either win the final game of the championship and earn the prize money or lose and earn nothing.

Outcomes are the potential results of an uncertain event. For example, in a basketball championship, the outcomes are either winning the game and the championship or losing both.

Payoffs represent the value associated with each outcome. Assume the team receives $10,000 if it wins and $6,000 if it loses. So, the payoffs in this example are $10,000 and $6,000.

Probability quantifies the likelihood of different outcomes in uncertain situations. For example, it is assumed that the college team has a 0.5 probability of winning the final game and a 0.5 probability of losing the final game.

Expected value is a useful tool for evaluating uncertain situations. It represents the average payoff across all possible outcomes, weighted by their probabilities. It is calculated by summing the products of each payoff and its associated probability. It is expressed as:

Expected Value = (P₁×X₁) + (P₂×X₂)

where P₁ and P₂ are the probabilities of the two outcomes, and X₁ and X₂ are their corresponding payoffs.

The expected income, which is the estimate of the college team’s expected earnings, is

(0.5×$10,000)+(0.5×$6,000) = $8,000

This analysis is helpful in decision-making under uncertainty.

People face uncertain situations. 

For example, Nicole’s manager informs her that she will receive a higher bonus if the company performs well and a lower bonus if its performance is average.

Outcomes are the possible results in uncertain situations. For Nicole, these are higher or lower bonuses depending on the company’s performance.

Payoffs represent the value associated with each outcome. For Nicole, the payoffs are $10,000 for a higher bonus and $5,000 for a lower bonus.

Probability is a measure of the likelihood that a particular outcome will occur in a situation of uncertainty. For example, the probability of the company performing well or on average is assumed to be 0.5 each.

The expected value is calculated by multiplying each payoff by its probability of occurring and then summing the weighted payoffs.

Here, the expected value is the product of a $10,000 payoff, and its 0.5 probability added to the product of a $5,000 payoff and its 0.5 probability, resulting in $7,500.

• Uncertainty - Situation wherein unknown future outcomes are majorly influenced by chance or exterior factors.
• Outcome - Possible result originated from uncertain situations.
• Payoff - Value attached to the outcomes.
• Probability - Statistical assessment of the likelihood of different consequences in uncertain conditions.
• Expected Value - Average payoff throughout all possible outcomes, addressed in ratio by their likelihood of occurrence.

• Define Uncertainty – Comprehend its occurrence through different scenarios (e.g., student's job chances post graduation).
• Contrast Outcome vs Payoff – Distinguish between the two in terms of value and the final result (e.g., championship game).
• Explore Examples – Demonstrate how uncertainty plays out in real-world scenarios (e.g., basketball championship).
• Explain Probability – Understand its role in determining outcomes.
• Apply Expected Value in Context – Describe how it helps in making decisions under uncertainty.

• What is uncertainty and how does it affect outcomes?
• How does a payoff differ from an outcome in an uncertain situation?
• How does probability impact the expected value?

• Students – Gain insight into how uncertainty plays out in various spheres of life.
• Educators – Offers a framework for making complex statistical concepts relatable.
• Researchers – Can assess decisions and behaviors under uncertainty.
• Decision Makers – Helps in decision-making process considering the various uncertain factors.