Confusing Conditional Probability Question

Hello all!

I'm trying to work through a problem involving conditional probability and I am more than confused. I need some guidance. Here is a brief description of the problem statement:

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Someone makes frequent airplane trips from LA to Wash. DC; She flies 50% of the time on airline #1, 30% on airline #2, and 20% on airline #3.

-For airline #1, flights are late into DC 30% of the time and late into LA 10% of the time.

-For airline #2, flights are late into DC 25% of the time and late into LA 20% of the time.

-For airline #3, flights are late into DC 40% of the time and late into LA 25% of the time.

Question: On a particular trip she arrived LATE at exactly 1 of 2 destinations. What are the posterior probabilites of having flown on airlines #1, #2, and #3?

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My professor hinted to use Baye's Theorem: P(A|B)P(B)=P(B|A)P(A).

So I know the basic question we want to ask is: What is the probablility that she flew #1 given she was late at one location? etc for #2 and #3.

I'll say: Event A=she flew #1, Event B=she flew #2, Event C=she flew #3. And say Event X=late at exactly 1 destination.

I want to compute: P(A|X), P(B|X), and P(C|X).

We know: P(A)=.5, P(B)=.3, and P(C)=.2

Now it is this event X that is getting me confused. Is X based on all 3 different situations? Or is X different depending on what airline we are looking at? I think it this problem can be done by looking at each airline seperately but I'm not really sure.

Any input would be much appreciated. I'll be working through it so I'll post any progress I make to see if I'm heading in the right direction.

Thanks!