Expected Values
Expected Values - Related Videos
01:15
Expected Value
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2023
The expected value is known as the "long-term" average or mean. This means that over the long term of experimenting over and over, you would expect this average. The expected average is represented by the symbol μ. It is calculated as follows:In the equation, x is an event, and P(x) is the probability of the event occurring.The expected value has practical applications in decision theory.This text is adapted from Openstax, Introductory Statistics, Section 4.2 Mean or Expected Value and...
01:19
Expected Income, Expected Utility, and Risk Aversion II
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2025
John is evaluating a job offer from a company where his income will be uncertain. If the company performs well, John will earn an annual income of $81,000; otherwise, he will earn $49,000. It is assumed that either outcome has an equal chance, assigning a probability of 0.5 to each. This results in an expected income of $65,000. His decision-making is affected by the diminishing marginal utility of income. John evaluates his options based on their utility. Expected utility accounts for risk...
01:24
Biodiversity and Human Values
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2019
Human civilization relies on biodiversity in many ways. Sudden changes in species biodiversity result in environmental changes that can modify weather patterns and therefore human civilizations. Humans are dependent on agriculture, which developed when ancestral humans found species that made suitable foods. At least 11,000 years ago, humans started to select plant and animal species to be cultivated on farms. Going back for thousands of years, humans have been artificially selecting species...
01:29
Professional Values
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2023
Nurses are responsible for caring for patients during birth, death, illness, and healing. Professional values guide the decisions and actions that nurses make in their careers. If nurses know the decisions and actions to take, providing patients with exceptional care is possible. The values that are the foundation of the nursing profession are altruism, autonomy, human dignity, and social justice. First, altruism refers to the concern for the welfare and well-being of others without personal...
01:08
Expected Income, Expected Utility, and Risk Aversion I
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2025
Consider a hypothetical example where John is evaluating a job offer from a company. If the company performs well, John will earn an annual income of $81,000; if it performs poorly, he will earn $49,000. Each outcome is equally likely, with a probability of 0.5. These two outcomes are mutually exclusive, meaning only one can occur and their probabilities sum to 1. The amounts of $81,000 and $49,000 represent the payoffs associated with each outcome.John's expected income is the average amount...
FAQs
For a discrete biological outcome, each possible result contributes according to its probability rather than being treated equally. The calculation multiplies each outcome by its probability and then sums those products. Highly probable outcomes therefore influence the long-run mean more strongly than rare outcomes, allowing a model to combine several possible results into one quantitative benchmark.
When outcomes vary along a continuous distribution, individual values are not simply listed and weighted one by one. Instead, the probability-weighted contribution is represented through integration over that distribution. This extends the same averaging principle to continuously varying biological measurements and produces a mean for interpreting uncertainty across the full range of possible values.
An expected value can fall between possible outcomes or otherwise fail to appear in any single observation. Its purpose is to summarize the long-run mean across repeated random outcomes, not to predict that every trial will equal the average. In biology, this distinction helps separate variation in individual results from the central quantity predicted by a probability model.
Comparing an expected value with an observed value tests how well a probability model represents biological data. A difference does not automatically invalidate the model, because observed outcomes can vary around the modeled expectation. The comparison becomes informative when used to assess model performance, interpret biological variation, or support quantitative risk assessment.
To estimate an inherited outcome, first specify the possible offspring traits and assign the inheritance probability to each one. Multiply every trait value by its corresponding probability, then add the products. The resulting expected value provides a probability-based estimate for the offspring traits, while individual offspring may still differ from that estimate.
In sampled biological populations, expected values provide a reference for quantities estimated from data. A researcher can compare the probability-based expectation with the measure obtained from a sample, then examine whether the difference reflects biological variation or a mismatch between the model and observations. This supports interpretation of population measures without treating one sample as the long-run result.
For stochastic biological experiments, the expected value supplies a benchmark before or alongside observing outcomes. It can summarize what the probability model predicts over repeated random results and allow comparison with experimental observations. This makes uncertainty measurable and helps frame biological risk assessment, even when no single trial equals the expected value.