The law of large numbers states that for independent repeated trials with a common success probability p, the observed success proportion approaches p as the number of trials grows. This is a long-run tendency specified by the law, meaning that while individual short-term outcomes may vary, the cumulative average stabilizes around the theoretical value over many repetitions.
Conditions: The trials are independent.; Each trial shares a common success probability p.; The number of trials grows large.
The law of large numbers states that for independent repeated trials with a common success probability p, the observed success proportion approaches p as the number of trials grows. This is a long-run tendency specified by the law, meaning that while individual short-term outcomes may vary, the cumulative average stabilizes around the theoretical value over many repetitions.
Conditions: The trials are independent.; Each trial shares a common success probability p.; The number of trials grows large.
The denominator P(B) in Bayes' theorem represents the total probability of the evidence (event B). In the context of a screening test, a positive result can arise from two mutually exclusive scenarios: a true positive (person has the disease and tests positive) or a false positive (person does not have the disease but tests positive).
Conditions: The evidence event B can occur via multiple mutually exclusive paths.; The law of total probability applies.
The denominator P(B) in Bayes' theorem represents the total probability of the evidence (event B). In the context of a screening test, a positive result can arise from two mutually exclusive scenarios: a true positive (person has the disease and tests positive) or a false positive (person does not have the disease but tests positive).
Conditions: The evidence event B can occur via multiple mutually exclusive paths.; The law of total probability applies.
In the geometric model, the total sample space is represented as a large square. The prior probability P(H) determines the width of a vertical strip representing the hypothesis, and the likelihood P(E∣H) determines the height of the shaded region within that strip where the evidence occurs.
Conditions: Total sample space is normalized to unit area; Events H and E are treated as measurable subsets
In the geometric model, the total sample space is represented as a large square. The prior probability P(H) determines the width of a vertical strip representing the hypothesis, and the likelihood P(E∣H) determines the height of the shaded region within that strip where the evidence occurs.
Conditions: Total sample space is normalized to unit area; Events H and E are treated as measurable subsets
Expected profit is a long-run average calculated by weighting outcomes by their probabilities. It does not guarantee that any single trial or even a finite series of trials will yield exactly that amount, because actual results vary randomly around the mean.
Conditions: The scenario involves random outcomes with assigned probabilities.; The calculation uses the probability-weighted average (expectation).
Expected profit is a long-run average calculated by weighting outcomes by their probabilities. It does not guarantee that any single trial or even a finite series of trials will yield exactly that amount, because actual results vary randomly around the mean.
Conditions: The scenario involves random outcomes with assigned probabilities.; The calculation uses the probability-weighted average (expectation).