Expected Probability Calculator
Expected Probability Calculator helps determine the expected probability of various outcomes in probability scenarios. It's essential for risk assessment, decision-making, and statistical analysis. Users can input multiple possible outcomes with their probabilities to calculate the expected value, which predicts average outcomes over time. Particularly useful in finance, insurance, and game theory for making informed decisions based on probabilistic outcomes.
Formula
Expected Probability (E) = Σ (Outcomei × Probabilityi)
Where:
E = Expected Value
Σ = Sum of all outcomes
i = Each individual outcome
Derivation Process
The expected value formula originates from probability theory's fundamental axioms. It combines each outcome's value with its likelihood, creating a weighted average. First conceptualized in 17th-century probability studies, it was formalized through Bernoulli's work on expectation. The derivation involves summing all possible outcomes multiplied by their respective probabilities, assuming exhaustive and mutually exclusive events. This mathematical expectation represents the long-run average outcome if an experiment were repeated infinitely.
FAQs
What is expected probability used for?
Expected probability calculates average outcomes in uncertain scenarios. It's crucial for decision-making in finance, insurance, and game theory, helping predict long-term results of repeated events by weighting outcomes by their likelihoods.
Can probabilities exceed 1 in calculation?
No, individual probabilities must be between 0 and 1, and total sum must equal 1. The calculator validates inputs to ensure mathematical integrity and accurate results.
How to handle percentage probabilities?
Convert percentages to decimals (e.g., 25% = 0.25) before input. The calculator uses decimal format for probabilities to maintain mathematical consistency with standard probability conventions.
What if probabilities don't sum to 1?
The calculator shows an error message. Valid probability scenarios require exhaustive outcomes where total probabilities exactly equal 1 for accurate expected value calculation.
Can I calculate negative outcomes?
Yes, outcomes can be negative numbers. This is common in financial calculations where losses (negative values) are possible outcomes in risk assessment models.
Is this applicable to stock market?
Absolutely. Investors use expected probability to evaluate potential returns by assigning probabilities to different market scenarios and calculating expected portfolio performance.
How accurate are the results?
Results are mathematically precise based on inputs. Accuracy depends on correct probability assignments and comprehensive outcome inclusion reflecting real-world possibilities.
Can I use fractions instead?
Yes, but convert to decimals first. The calculator accepts decimal inputs (e.g., 1/3 = 0.3333). Maintain sufficient decimal places for precision in calculations.
What's the difference from average?
Expected value weights outcomes by probability, unlike simple average. It's probability-weighted mean rather than arithmetic mean, making it more relevant for uncertain outcomes.
Why see multiple decimal places?
The calculator shows precise results with multiple decimals to maintain accuracy. Users can round final results as needed for their specific application requirements.