On a given day, machine a has a 10% chance of malfunctioning and machine b has a 7% chance of the same. given that at least one of the machines malfunctioned today, what is the chance that machine b malfunctioned?

I will give an instance to answer this. Suppose that we have 1,000 days. And it was given above that machine A has 10 % chances to malfunctioned and B has a 7%.  So machine A has 100 days that malfunctioned while machine B has 70 days. And 7 days that both machines malfunctioned (10%x7%). Next is to add 100 days from machine A and 70 days from machine B and subtract with the 7 days where both of them malfunctioned. So the result is 163 days out of 1000 that we expect an incidence of failure. So the chance of machine B to malfunctioned is 70 days. 70/163 is 42.93%