Brand new to this site. I've read the posting rules, but please feel free to herd me into the correct way of doing things with this community.

I am dealing with this problem below. If you can see where my thinking is going awry, feel free to talk to me like a little kid. I want to understand more than I want an answer.

Problem

At a certain gas station, 40% of all customers fill their tanks. Of those who fill their tanks, 80% pay with a credit card. If three customers are randomly selected, what is the probability that all three fill their tanks and pay with a credit card?

My Work

Translation: P(Credit | Fill) = .8 Which means that P(Cash | Fill) = .2.

This seems like I should be finding a mixture of joint and compound probabilities. From the question, I think I want to find "P(C and F) and P(C and F) and P(C and F)."

P(C and F) should be 0.32:

P(C|F) = P(F and C) / P(F)

0.8 = P (F and C) / .4

0.32 = P(F and C)

32%

So to solve, I do this:

P(C and F) and P(C and F) and P(C and F)

= P(C and F) * P(C and F) * P(C and F)

= .32 * .32 * .32

= 3.28 %

The comment my professor left me was: Why are you multiplying them, not adding them here?

I have been all over youtube, and all over several stats sites, and while I can maybe see where I should be doing something more like, P(C|F) and P(C|F) and P(C|F), but the question isn't asking that. There are people who fill their tanks and pay with cash/checks. Can you help me figure out where my thinking went awry?

I feel like if I did what my prof told me and added them, I would get an incredibly high number (96%), but also I would be answering the question, "what is the probability of choosing three random cars where the first or the second or the third paid with credit card?"

Any help is appreciated - I hope I have shown my work without being too wordy or aggravating.

Thanks,

Dani