A hairdresser sets up shop in the central business district. All her customers are bankers or lawyers, with 70% of them being bankers and 30% lawyers. The wicked hairdresser enjoys dyeing the customers’ hair pink and pretending it was an accident. She dyes a banker’s hair pink with probability 0.2, and a lawyer’s with probability 0.4.

The sample space is Ω = {customers}. Define the events B = {customer is a banker}, L = {customer is a lawyer}, P = {customer’s hair is dyed pink}.

a. What’s the probability a customer gets their hair dyed pink?

b. An unlucky customer is seen leaving the hairdresser’s shop with pink hair. What’s the probability she is a lawyer?

c. Sometimes customers sue the hairdresser after a misadventure with pink hair. A banker with pink hair will sue her with probability 0.02, while a lawyer with pink hair will sue her with probability 0.1. The hairdresser is never sued by customers without pink hair. Define the event S = {customer sues the hairdresser}.
Write down the three pieces of information above as probability statements. Hence find P(S), the probability that a customer sues the hairdresser.