To find the probability of hanging the blue painting first and then the green painting second, we can analyze the situation step by step.
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Probability of hanging the blue painting first: There are 5 paintings and only 1 of them is blue. Thus, the probability of picking the blue painting first is:
\[ P(\text{blue first}) = \frac{1}{5} \]
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Probability of hanging the green painting second given that the blue is first: After hanging the blue painting, there are 4 paintings left, and only 1 of them is green. The probability of picking the green painting second is:
\[ P(\text{green second | blue first}) = \frac{1}{4} \]
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Final probability of both events happening: To find the combined probability of both events (hanging blue first and green second), we multiply the individual probabilities:
\[ P(\text{blue first and green second}) = P(\text{blue first}) \times P(\text{green second | blue first}) = \frac{1}{5} \times \frac{1}{4} = \frac{1}{20} \]
Thus, the probability of hanging the blue painting first and then the green painting second is:
\[ \boxed{\frac{1}{20}} \]