There are two neighboring towns, Springfield and Shelbyville, and these towns have a raffle every year at the big Springfield vs. Shelbyville football game. The prize money is split between two winners whose tickets match the ones drawn randomly. The only condition is that they each must donate half of their prize money to one of two charities, but they may select which one. The two charities are The Human Fund (THF) and The Yzma Kronk Right Lever Foundation (YK). The organizers of the event are interested in the likelihood of where the charitable donation will end up and which town the winner will be from, so before the game they take a poll of the townsfolk in each town and assess which charity they support. 70% of Springfielders support THF, while only 40% of Shelbyvillians support THF. The remaining portion of each town supports YK.
Two winners are identified, and paid their prize money. The event organizers know that 60% of the tickets went to Springfielders and 40% were sold to Shelbyvillians, but they don’t know which town either of the two individual winners are from.
Using the information provided below, please answer the questions.
a. The first winner donates half of her prize money to THF. Given this information, what is the probability that the first winner was from Springfield? What is the probability she hailed from Shelbyville instead?
b. The second winner donates half of his prize money to the YK. Given this information, and the knowledge from part a above, what is the probability that he is from Springfield? From Shelbyville?