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The question is:

A machine produces small cans that are used for baked beans. The probability that the can is in perfect shape is 0.9. The probability of the can having an unnoticeable dent is 0.02. The probability that the can is obviously dented is 0.08. Produced cans get passed through an automatic inspection machine, which is able to detect obviously dented cans and discard them. What is the probability that a can that gets shipped for use will be of perfect shape?

I can't seem to figure this one out. I haven't looked at the answer yet because I don't want to work backwards and I am wondering if anyone can just give me a hint on what I'm suppose to do? I need to get this stuff down for the P exam, I hope to study hard for the next 2 months and take it over the summer.

Here's my thought process: Let P(A) = perfect, P(B) = Shipped. P(A) = .9 and we want to find P(B|A). I know this is $P(B\,intersect\,A)\over P(A)$. I can't seem to figure out what B intersect A is.

Edit: With help from Pepe, I need to find P(A|B) and I can recognize P(A) = .9 and P(B) = .92. Also, P(B|A) = 1 since it will always ship if it is perfect so using this I was able to find that P(B intersect A) is .9. Then I simply did .9/.92 to get .978. If anyone would be willing to check and see if this is right, I would appreciate it.

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    First notice that the question asks you to find $\mathbb{P}[A|B]$, and $\mathbb{P}[B|A]=1$. You know $\mathbb{P}[A]$ and $\mathbb{P}[B]=0.9$, do you think you could finish it from there? – 2017-02-23
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    @PepeSilvia How come P(B) = .9 as well? I think I can figure it out from here but I don't see why P(B) =.9. Also, thank you for pointing out that it is A|B, I didn't see that. Edit: Wouldn't P(B) = .92? – 2017-02-23
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    Yes sorry, $\mathbb{P}[B]=0.92$!. – 2017-02-23

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Tip: You want the probability that the can is perfectly shaped given that it is not obviously dented.

You have the probabilities for each of three mutually exclusive and exhaustive events: $S, U, N$, for Shapely, Unnoticeably dented and Noticeably dented.

Given: $\mathsf P(S)=0.9, \mathsf P(U)=0.02, \mathsf P(N)=0.08$, mutual exclusion, and exhaustion,

Find: $\mathsf P(S\mid S\cup U)$

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    Correct me if I'm wrong, but doesn't this lead to the answer the OP has arrived at? – 2017-02-23
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    I think I just wrote mine a little differently than you, but in my edit I had B instead of S$\cup$U and S$\cup$U is just .9+.02 or so I thought, which is what B is also equal to. – 2017-02-23
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    It does now the edit shows up. – 2017-02-23