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Home Depot carries 3 brands of paint. A customer wants to buy another gallon of paint to match paint that she bought at the store previously. She can't recall the brand name and does not wish to return home to find the old can of paint and therefore she selects two brands of paint at random and buys them. What is the likelihood that she has chosen the right brand?

I'm confused between two possible answers:
I think it should be $\frac{5}{9}$ by finding the complement of buying neither correct which would be calculated by $1-(\frac{2}{3}\times\frac{2}{3})$.
Another possible answer is $\frac{2}{3}$ which I don't think is correct and not sure how to get.

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  • $\begingroup$ Welcome to Math SE. Hint: Is each of the $2$ cans of paint not being correct completely independent of each other? $\endgroup$ – John Omielan Oct 2 '19 at 5:35
  • $\begingroup$ From a selection of a Jack, Queen, and King from a deck of cards, what is the probability of having a Queen if you can select two cards at random? Play out the scenario and you'll see that the answer is $\frac23$. $\endgroup$ – Andrew Chin Oct 2 '19 at 5:37
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I think it should be 2/3. The sample space looks like:

{B,Y}, {R,B}, {R,Y}

assuming the colours are R = red, B = blue, Y = yellow. And assuming also she won't pick something like {B,B}. If we want we can have them in the sample space and assign them probability value 0.

We can assign each of our events the probability value 1/3. And if one of the allocated colors is correct, it would appear in 2 events of the sample space. This results in the probability of 2/3.

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    $\begingroup$ The question statement of "... she selects two brands of paint ..." (emphasis mine) makes it very clear, at least to me, she won't pick something like {$B,B$}. $\endgroup$ – John Omielan Oct 2 '19 at 6:00
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    $\begingroup$ You are correct. I sometimes miss out on the subtle details of these questions. $\endgroup$ – tisPrimeTime Oct 2 '19 at 6:01
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I was earlier considering them to be independent. But she won't choose the same two - they are not independent - and the answer is indeed $\frac{2}{3}$.

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  • $\begingroup$ It would be better for you to make this statement a comment to your first answer, or edit that answer, rather than opening another answer. The only time when it's appropriate to open another answer is when you actually have another, independent answer, e.g. a solution by a totally different method--and even then it may be better to combine both solutions into a single answer. $\endgroup$ – awkward Oct 2 '19 at 12:53

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