To find the probability of hanging the blue painting first and then the green painting second, we can break down the problem into two steps:
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Probability of hanging the blue painting first: Since there are 5 paintings in total, and only one of them is blue, the probability of selecting the blue painting first is: \[ P(\text{Blue first}) = \frac{1}{5} \]
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Probability of hanging the green painting second after the blue: Once the blue painting has been hung first, there are 4 paintings left (red, yellow, green, and purple). Out of these, only one is green. Therefore, the probability of selecting the green painting as the second is: \[ P(\text{Green second | Blue first}) = \frac{1}{4} \]
Now, to find the total probability of both events happening (hanging blue first, then green second), we multiply the probabilities of the two events:
\[ P(\text{Blue first and Green second}) = P(\text{Blue first}) \times P(\text{Green second | Blue first}) = \frac{1}{5} \times \frac{1}{4} = \frac{1}{20} \]
Therefore, the probability of hanging the blue painting first and then the green painting second is:
\[ \frac{1}{20} \]
However, none of the provided options (A, B, C, D) matches this. Let's go through the options again:
- A. \( \frac{1}{5} \)
- B. \( \frac{1}{5} \cdot \frac{1}{5} \)
- C. \( 1.5 \cdot \frac{1}{4} \)
- D. \( \frac{1}{5} + \frac{1}{5} \)
While we were looking for \( \frac{1}{20} \), the answer doesn't seem to appear in the options provided. However, from our calculation, we have determined that the actual probability of the scenario described (blue first, green second) is \( \frac{1}{20} \).
If interpreting the options strictly:
- Option B is \( \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{25} \), which is incorrect.
- Option C calculates to \( \frac{1.5}{4} = \frac{3}{8} \), which is incorrect.
- Option D sums to \( \frac{2}{5} \), also incorrect.
Thus, the options provided do not encapsulate our calculated answer.
The correct probability of hanging the blue painting first and then the green painting second is indeed \( \frac{1}{20} \).