Thank you JeffM for your help, it could be more helpful for me if you please solve this as i'm totally new to Probability and learning through examples.
OK Let's go through the simpler problem. Then you try the original problem and show your work. That way we can see where you are going wrong if you are.
If 23% of the customers pay in full each month, then the probability of selecting such a customer if you choose one customer at random is 23%. Does that make sense?
American Express has millions of customers so the probabilities scarcely change if we choose a small sample at random. Does that make sense?
Suppose we choose three customers at random. What is the probability that all three pay their bills in full each month?
Jump to theory. The probability of a compound event, say A and B, follows this rule
\(\displaystyle P(A\ and\ B) = P(A\ if\ B) * P(B) = P(B\ if\ A) * p(A).\) This is one of the fundamental theorems of probability.
Another fundamental theorem is this: \(\displaystyle A\ and\ B\ are\ independent \implies P(A\ if\ B) = P(A)\ and\ P(B\ if\ A) = P(A)\ and\ vice\ versa.\)
For all practical purposes, the probability of choosing American Express customers with a given characteristic in a small sample is independent.
Putting those two theorems together we get:\(\displaystyle A\ and\ B\ are\ independent \implies P(A\ and\ B) = P(A\ if\ B) * P(B) = P(A) * P(B).\)
So the probability of getting three customers who all pay their bills in full each month = 0.23 * 0.23 * 0.23 = 0.012167 = 1.2167%. With me so far?
The probability of getting three customers who all do not pay their bills in full each month = 0.77 * 0.77 * 0.77 = 0.456533 = 45.6533%. With me so far?
The probability of getting one customer who does pay the bill in full each month and two who do not is harder to figure out. The problem is that there are really three possibilities, the first person selected is the one who pays in full, or the second person selected is the one who pays in full, or the third person selected is the one who pays in full. So the probability = 3 * 0.77 * 0.77 * 0.23 = 0.409101 = 40.9101%. Do you follow that? That is the key concept.
The probability of getting two customer who pay their bills in full each month and one who does not follows similar logic. The probability is
3 * 0.77 * 0.23 * 0.23 = 0.122199 = 12.2199%.
Notice 1.2167 + 45.6533 + 40.9101 + 12.2199 = 100.0000. That is a check that we did the process correctly.
Did you understand that? If not, please tell me where I lost you.