A company has two types of machine that run 24 hours a day, 7 days a week. Type A machine breaks down on average 10 breakdowns per day. This is equivalent to saying the mean wait time per breakdown is days per breakdown. For type B machine, the mean wait time between breakdowns is 0.7 days per breakdown.
For each type of machine, assume breakdowns occur as a time homogeneous Poisson process, so that for each type of machine the waiting time for a breakdown has exponential distribution, but with differing parameters depending on whether the machine is type A or type B.
Suppose 25% of the machines at this company are type A.
You observe a machine but cannot tell what type it is. But you do observe that within the next 5 hours it does not break down. What is the posterior probability this is a type A machine? Answer as a decimal accurate to 4 decimal places.