A store analyzed a sample of its recent sales and recorded the value of the sale and the method of payment. The results are summarized in the following table:

Show basic workings. Answers may be left as fractions.

a) What is the probability that a customer used cash?

b) What is the probability that a cash purchase was under $20?

c) What is the probability that a customer made a purchase under $20 or used cash?

d) What is the probability that a customer paid by cash, given that they spent under $20?

e) Are the events "payment by cash" and "purchase under $20" independent? Explain.