Painting Probability Problem

To solve this problem we need to use the Bayes' Theorem. Let’s denote the following events:

- $A$: The painting is a copy.
- $B$: The person believes the painting is original.

We want to find the probability that the painting is actually a copy given that the person believes it is original. In Bayesian terms, this is $P(A \mid B)$.

We start by using Bayes' Theorem:
\[P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)} \]

Note that: 


- The probability that a painting is a copy is $P(A) = 0.25$.
- Consequently, the probability that a painting is original is $P(A^c) = 1 - P(A) = 0.75$.


- The person makes a mistake 20% of the time. Thus, if a painting is a copy, there is an 80% chance the person will incorrectly identify it as original: $P(B \mid A) = 0.20$.
- If a painting is original, the probability that the person correctly identifies it as original is 80% (so there's a 20% chance of mistakenly identifying it as a copy):  $P(B \mid A^c) = 0.80$.

Calculate the total probability that the person believes the painting is original ($P(B)$):
\[ P(B) = P(B \mid A) \cdot P(A) + P(B \mid A^c) \cdot P(A^c) \]
\[ P(B) = (0.20 \cdot 0.25) + (0.80 \cdot 0.75)=0.65\]

Now we have
\[P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)} \]
\[ P(A \mid B) = \frac{0.20 \cdot 0.25}{0.65}=0.0769=7.69\%\]


Therefore, the probability that the painting is actually a copy given that the person believes it is original is approximately $7.69\%$.