Suppose you arrange 5 marbles on a table for decoration from left to right. One marble is yellow, one is blue, one is red, one is green, and one is orange. What is the probability that you arrange the marbles so that the green one is first on the left?

To find the probability of arranging the marbles such that the green one is first on the left, we need to determine the total number of possible arrangements and the number of favorable arrangements where the green marble is first.

Step 1: Determine the total number of possible arrangements.
Since there are 5 marbles in total, there are 5 possible choices for the first marble, 4 remaining choices for the second marble, 3 remaining choices for the third marble, 2 remaining choices for the fourth marble, and 1 remaining choice for the last marble. Thus, the total number of possible arrangements is given by the factorial of 5:
$$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$

Step 2: Determine the number of favorable arrangements where the green marble is first.
Since we want the green marble to be first, we fix its position on the left. We then have 4 remaining marbles to arrange, namely the yellow, blue, red, and orange marbles. There are 4 possible choices for the second position, 3 remaining choices for the third position, 2 remaining choices for the fourth position, and 1 remaining choice for the last position. Thus, the number of favorable arrangements is given by the factorial of 4:
$$4! = 4 \times 3 \times 2 \times 1 = 24$$

Step 3: Calculate the probability.
The probability of arranging the marbles such that the green one is first is given by the number of favorable arrangements divided by the total number of possible arrangements:
$$\text{Probability} = \frac{\text{Number of Favorable Arrangements}}{\text{Total Number of Possible Arrangements}} = \frac{24}{120} = \frac{1}{5}$$

Answer: The probability of arranging the marbles such that the green one is first on the left is $\frac{1}{5}$.