Why does swapping the conditioning event and the target event usually change the conditional probability?

Starting from the equality $P(A)P$(B|A) = $P(B)P$(A|B), you can solve for either conditional probability by dividing by the corresponding marginal probability. Dividing both sides by $P(B)$ isolates P(A|B), giving P(A|B) = $P(A)P$(B|A)/P(B).

Conditions: Both A and B have positive probability for the ordinary conditionals used here.; The algebraic rearrangement requires the denominators $P(B)$ and $P(A)$ to be nonzero.

The values $P(A\cap B)=0.0225$ and $P(B)=0.5$ are described as illustrative inputs because they are hypothetical numbers chosen to demonstrate the calculation method, rather than verified population statistics. This distinction prevents learners from misinterpreting the example as factual demographic data.

Conditions: The context is a pedagogical example in a video.; The events involve alcohol dependence and gender.

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 $1\%$.

The event 'A and B' is logically identical to 'B and A', so any valid decomposition of its probability must agree. This symmetry forces the equality of the two product formulas: $P(A)P$(B|A) = $P(B)P$(A|B).

Conditions: The argument uses commutativity of logical conjunction for events.; Both A and B have positive probability for the ordinary conditionals used here.

Using the assumed values from the classroom example where $A$ denotes alcohol dependence and $B$ denotes being a man, you divide the probability of the intersection $P(A\cap B)$ by the probability of the conditioning event $P(B)$. With $P(A\cap B)=0.0225$ and $P(B)=0.5$, the calculation is $0.0225/0.5=0.045$, or $4.5\%$.

Conditions: $P(A\cap B) = 0.0225$.; $P(B) = 0.5$.; These are illustrative inputs, not verified population statistics.