Should it be A|B or B|A?

Question

I came across a question in condition probability which left me wondering.

A departmental store reports that 30% of sales are transacted via cash, 60% are credit card and 10% with a debit card.

20% of the above cash purchases, 90% of the above credit card purchases and 80% of the above debit card purchases are made for purchases more than $200.

So Let $C = {cash}$, D = {credit card}, E = {debit card} and M = {purchases > $200}

P(C) = 0.3
P(D) = 0.6
P(E) = 0.1

What I am not really sure is if it should be $P(M|C) = 0.2$ or is it $P(C|M) = 0.2$. Was wondering if I may get some advice on this please.

Answer 1

$P(M|C)$ denotes probability of the purchase being above $200 when you have observed that the purchase is through cash.

You say 20% of the cash purchases are made for above $200$. So $P(M|C) = 0.2$

$P(C|M)$ is slightly more complicated. You observe that the purchase is above $200 and then you need the conditional probability that it is done by paying cash. You need to apply Bayes rule.

$$ P(C|M) = \frac{P(C ) P(M|C)}{P(M)} = \frac{0.3 \times 0.2}{0.68} \approx 0.088 $$

P.S: I must add, I am a newbie myself. So please get this verified by someone!

P.P.S: Sorry, as pointed out below, I made a small error with my normaliser. Fixed now.