To find the probability of Bathsheba hanging the blue painting first and then the green painting second, we need to determine the total number of ways to arrange the 5 paintings and the specific arrangements that meet our criteria.
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Total arrangements of 5 paintings:
The total number of ways to arrange 5 distinct paintings is calculated using the factorial of the number of paintings: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] -
Favorable arrangements (blue first, green second):
If we fix the blue painting in the first position and the green painting in the second position, we now have 3 remaining paintings (red, yellow, and purple) to arrange in the last three positions. The number of ways to arrange these 3 paintings is: \[ 3! = 3 \times 2 \times 1 = 6 \] -
Calculating the probability:
The probability of the event that the blue painting is first and the green painting is second is the ratio of the number of favorable arrangements to the total arrangements: \[ \text{Probability} = \frac{\text{Number of favorable arrangements}}{\text{Total arrangements}} = \frac{3!}{5!} = \frac{6}{120} = \frac{1}{20} \]
Therefore, the probability of hanging the blue painting first and then the green painting is: \[ \boxed{\frac{1}{20}} \]