Conditional Probability Explained: Visual Intuition

The model illustrates the difference by showing that $P(+\mid D)$ (sensitivity) is fixed at $95\%$, while $P(D\mid +)$ (positive predictive value) is calculated to be only $~16.1\%$. This demonstrates that the probability of a positive test given disease is not the same as the probability of disease given a positive test, especially when prevalence is low.

Conditions: Sensitivity is $95\%$.; Calculated $P(D\mid +)$ is $~16.1\%$.; Prevalence is $1\%$.

False positives are more numerous because the healthy population is much larger than the diseased population. Even though the false positive rate is low ($5\%$), applying it to a large base (1,980 people) yields a higher absolute count (99) than applying the high sensitivity rate ($95\%$) to a small base (20 people), which yields 19.

Conditions: Prevalence is $1\%$ (20 diseased, 1,980 healthy).; False positive rate is $5\%$.; Sensitivity is $95\%$.

A positive test result does not imply a high probability of illness because the base rate (prevalence) of the disease is very low compared to the healthy population. Even with high sensitivity and specificity, the sheer number of healthy individuals produces a large absolute number of false positives.

Conditions: The disease prevalence is $1\%$.; The test has $95\%$ sensitivity and $95\%$ specificity.; The population is a hypothetical model of 2,000 people.

Prevalence determines the size of the diseased and healthy subgroups in the population. It acts as the prior probability that influences the final posterior probability of disease given a positive test.

Conditions: The model uses a fictional population.; Prevalence is set to $1\%$.

Sensitivity is the probability of a positive test result given that the person has the disease ($P(+\mid D)$), while specificity is the probability of a negative test result given that the person does not have the disease ($P(-\mid D^c)$). In the model, both are assumed to be $95\%$, but they apply to different subgroups: the sick versus the healthy.

Conditions: The test has $95\%$ sensitivity.; The test has $95\%$ specificity.; The model distinguishes between diseased and non-diseased populations.

The model produces 19 true positives and 99 false positives. True positives are calculated from the 20 diseased individuals using the $95\%$ sensitivity, while false positives are calculated from the 1,980 healthy individuals using the $5\%$ false positive rate (derived from $95\%$ specificity).

Conditions: Population size is 2,000.; Prevalence is $1\%$ (20 diseased, 1,980 healthy).; Sensitivity is $95\%$.; Specificity is $95\%$.

To calculate the probability of having the disease given a positive test result, you must count the number of true positives and divide it by the total number of positive results (true positives plus false positives). In the model, this involves identifying the 19 true positives out of the 118 total positive tests, yielding approximately $16.1\%$.

Conditions: Prevalence is $1\%$ in a population of 2,000.; Sensitivity is $95\%$.; Specificity is $95\%$.

The base-rate fallacy occurs because learners often ignore the low prevalence of the disease (the base rate) and focus only on the high accuracy of the test. This leads to the incorrect assumption that a positive test result is highly likely to be a true positive, when in fact the large number of healthy people generates many more false positives than true positives.

Conditions: Prevalence is low ($1\%$).; Test accuracy (sensitivity/specificity) is high ($95\%$).; The observer focuses on test accuracy rather than population composition.