# What is this conditional probability asking?

I really dont understand how this info helps me solve this problem: In a study it was discovered that 25% of the paintings of a certain gallery are not original. A collector in 15% of the cases makes a mistake in judging if a painting is authentic or a copy. If she buys a piece thinking that it is original, what is the probability that it is not?

Now my question is how does any of this info help solve the problem here?

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We interpret the information we were given about the collector to mean: (a) if she thinks a painting is an original, then with probability $0.15$ she will be mistaken (so with probability $0.85$ she will be right) and (b) if she thinks it is a copy, then with probability $0.15$ she will be mistaken.

Let $T$ be the event that the collector thinks a painting is original, and let $C$ be the event that a painting is a copy. We want $\Pr(C|T)$. By the usual formula for conditional probabilities, we have $$\Pr(C|T)=\frac{\Pr(C\cap T)}{\Pr(T)}.\tag{1}$$ Now we need to find the probabilities on the right of Formula $(1)$.

The event $T$ can happen in two ways (i) the painting is a copy, and the collector thinks it is original or (ii) the painting is an original, and the collector thinks it is original.

To find the probability of (i), note that the probability the painting is a copy is $0.25$. Given that the painting is a copy, the probability the collector thinks it is original is $0.15$. so the probability of (i) is $(0.25)(0.15)$.

For the probability of (ii), note that with probability $0.75$ the painting is original. Given that it is original, the probability the collector thinks it is original is $0.85$. So the probability of (ii) is $(0.75)(0.85)$. It follows that $$\Pr(T)=(0.25)(0.15)+(0.75)(0.85).$$

For $\Pr(C\cap T)$, note that we have already computed it, it is just the probability of (i), which is $(0.25)(0.15)$.

Now use Formula $(1)$. As a check on your calculations (or mine), the answer turns out to be $\dfrac{1}{18}$.

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Thanks so much that really helps a lot – TheHopefulActuary Oct 9 '12 at 17:35

Hint

Denote $F$ if the portrait is fake and $O$ if the portrait is original. We must have $O=F^c$, as if the portrait isn't fake, it is original. Denote $C$ if the collector thinks the portrait is original. Then $C^c$ is that the collector thinks it is a fake.

From the question $Pr(F)=0.25$, $Pr(C|F)=0.15$ and $Pr(C^c|O)=0.15$.

You want $Pr(O|C)$, which is the probability it is original, given she thinks it is original.

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