let E be non empty finite set ; :: thesis: for A, B1, B2, B3 being Event of E st 0 < prob B1 & 0 < prob B2 & 0 < prob B3 & (B1 \/ B2) \/ B3 = E & B1 misses B2 & B1 misses B3 & B2 misses B3 holds
prob (B1,A) = ((prob (A,B1)) * (prob B1)) / ((((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))) + ((prob (A,B3)) * (prob B3)))
let A, B1, B2, B3 be Event of E; :: thesis: ( 0 < prob B1 & 0 < prob B2 & 0 < prob B3 & (B1 \/ B2) \/ B3 = E & B1 misses B2 & B1 misses B3 & B2 misses B3 implies prob (B1,A) = ((prob (A,B1)) * (prob B1)) / ((((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))) + ((prob (A,B3)) * (prob B3))) )
assume that
A1: 0 < prob B1 and
A2: ( 0 < prob B2 & 0 < prob B3 & (B1 \/ B2) \/ B3 = E & B1 misses B2 & B1 misses B3 & B2 misses B3 ) ; :: thesis: prob (B1,A) = ((prob (A,B1)) * (prob B1)) / ((((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))) + ((prob (A,B3)) * (prob B3)))
prob A = (((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))) + ((prob (A,B3)) * (prob B3)) by A1, A2, Th51;
hence prob (B1,A) = ((prob (A,B1)) * (prob B1)) / ((((prob (A,B1)) * (prob B1)) + ((prob (A,B2)) * (prob B2))) + ((prob (A,B3)) * (prob B3))) by A1, XCMPLX_1:87; :: thesis: verum