At a certain gas station, 55% of the customers purchase regular unleaded gas, 30% purchase super unleaded gas and the rest purchase premium unleaded gas. Of those customers purchasing regular gas, 70% fill their gas tanks (i.e. buy enough to bring the gas tank to capacity as opposed to buying a lesser amount of gas). Among customers purchasing super unleaded, 62% fill their tanks, while for purchasers of premium, 50% fill their gas tank.

Use R to denote regular unleaded gas, S for super unleaded and P for premium gas. Let F be the event that a customer fills his/her gas tank on a visit to the gas station.

a) What is the probability that a customer will fill his/her gas tank?

b) What is the probability that a customer purchased premium gas but did not fill the tank?

c) If the next customer at the gas station fills his tank, what is the probability that he purchased super unleaded gas?

d) Given that a customer did not purchase premium gas, what is the probability that he did not fill his tank?

e) Is type of gas purchased independent of whether a customer fills his tank? Prove your answer mathematically.