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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.

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 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.

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.

Substitute the given numerical values into the rearranged formula P(B|A) = $P(B)P$(A|B)/P(A). For example, if $P(B) = 1/21$, P(A|B) = $4/10$, and $P(A) = 24/210$, the calculation is ($1/21 * 4/10$) / ($24/210$).

Conditions: The formula used is the rearranged Bayes identity for P(B|A).; The given probabilities must be consistent with the formula's assumptions (e.g., $P(A) > 0$).

Substitute the given numerical values into the rearranged formula P(B|A) = $P(B)P$(A|B)/P(A). For example, if $P(B) = 1/21$, P(A|B) = $4/10$, and $P(A) = 24/210$, the calculation is ($1/21 * 4/10$) / ($24/210$).

Conditions: The formula used is the rearranged Bayes identity for P(B|A).; The given probabilities must be consistent with the formula's assumptions (e.g., $P(A) > 0$).