There are three underlying assets in an investment pool. Each has a 6% chance of defaulting. If an asset does not default it pays $1. If it defaults, it pays $0. The first two assets have a 40% correlation between each other. The third asset is uncorrelated with the first 2.
Financial engineers create three products. They have payoffs as follows:
Product 1: Pays $1 if any of the three assets pays, 0 otherwise.
Product 2: Pays $1 if at least two of the three assets pay, 0 otherwise.
Product 3: Pays $1 if all three of the assets pay, 0 otherwise.
What is the probability of each asset paying off?
Please show all of your work