A brokerage firm has two types of clients, those who prefer high-risk portfolios and those who prefer low-risk portfolios. Based on past data, it is found that 20% of clients prefer high-risk portfolios. Of the clients who prefer high-risk portfolios, 75% also incorporate some form of leverage into their portfolios (e.g., make their portfolio even riskier by investing with some debt). Clients who prefer low-risk portfolios, however, are much less likely to use leverage: only 20% of low-risk portfolios will incorporate leverage at all.

a) What is the probability that a randomly selected client will use leverage in their portfolio?

b) What is the probability that a randomly selected client won't use leverage in their portfolio?

c) Suppose that a client decides to not use leverage. What is the probability that this client plans to invest in a low-risk portfolio? d) It is further given that 60% of the clients who use leverage will have some financial issue causing them to liquidate their portfolio within the year. What is the probability that a client will use leverage and liquidate their portfolio?