(Monty Hall) In a game u r offered 3 doors, 2 empty, 1 with $100K behind. U chose Door 3 and Host open door 2 showing empty. Should you switch to Door 1?

Answer

P(B1|H2) =P(B1^H2)/P(h2)=P(H2|B1)*P(B1)/P(H2)=1*0.3333/0.5=2/3 >1/3 (H=Host Open i b=behind i)


P(H2) = Partition Sum =P(H2|B1)*P(B1)+P(H2|B2)*P(B2)+P(H2|B3)*P(B3)

Host knows not to open Door 1 so P(H2|B1)=1, P(H2|B2)=0
Host knows safe to open either door 1 or 3 so P(H2|B3)=1/2 to be fair.
So P(H2)=1*1/3 +0*1/3 + 1/2*1/3=1/2


Some basics:
P(AUB) +P(A^B)=P(A)+P(B)
P(A|B) = P(A^B)/P(B) by definition
P(A) = Sum of P(A|pi)*P(pi)  where {Pi} is a partition.