Given:
P(A∣B)=P(B∣A) and A∩B=ϕ
Since A∩B=ϕ, events A and B can occur together, meaning P(A∩B)>0.
The conditional probability formulas are:
P(A∣B)=P(B)P(A∩B)
P(B∣A)=P(A)P(A∩B)
Using the given condition P(A∣B)=P(B∣A):
P(B)P(A∩B)=P(A)P(A∩B)
Cross-multiplying:
P(A∩B)⋅P(A)=P(A∩B)⋅P(B)
Since P(A∩B)>0, dividing both sides by P(A∩B):
P(A)=P(B)
Therefore, P(A)=P(B).