Deriving Bayes’ Theorem: How Joint Probability Symmetry Reveals Conditional Reversal

Consider two boxes, A and B, each containing four balls. Box A holds three red and one green ball, while box B holds one red and three green balls. An eye‑blind person randomly selects a box with probability 1/2 and then draws a ball.

Simple probabilities are P(A)=P(B)=1/2. Conditional probabilities are derived directly from the composition of each box:

P(R | A) = 3/4 = count of red balls in A / total balls in A
P(G | A) = 1/4
P(R | B) = 1/4
P(G | B) = 3/4

Joint probabilities combine the box‑selection and ball‑draw steps:

P(R ∩ A) = P(A)·P(R | A) = 1/2·3/4 = 3/8
P(G ∩ A) = 1/2·1/4 = 1/8
P(R ∩ B) = 1/2·1/4 = 1/8
P(G ∩ B) = 1/2·3/4 = 3/8

These four non‑overlapping blocks partition the entire sample space, as confirmed by the sum:

P(R ∩ A) + P(G ∩ A) + P(R ∩ B) + P(G ∩ B) = 3/8 + 1/8 + 1/8 + 3/8 = 1

To answer the reversed question—"Given a red ball, what is the probability it came from box A?"—first compute the marginal probability of drawing a red ball:

P(R) = P(R ∩ A) + P(R ∩ B) = 3/8 + 1/8 = 1/2

Then apply the definition of conditional probability:

P(A | R) = P(R ∩ A) / P(R) = (3/8) / (1/2) = 3/4

The same reasoning for a green ball yields P(B | G) = 3/4.

This perspective shift—from conditioning on the box to conditioning on the ball—demonstrates the essence of Bayes’ theorem. The general formula emerges naturally:

P(A | R) = P(R | A)·P(A) / P(R)

Thus, Bayes’ theorem is simply a consistent way to re‑partition the joint probability space and invert the direction of conditioning.