To calculate the probability that Julio will be assigned the same color in both sessions, we first need to consider the total number of different outcomes for both sessions.
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In the first session, Julio can be assigned to one of three colors: Red, Yellow, or Green. This gives him 3 possible choices.
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In the second session, regardless of his assignment in the first session, he can also be assigned to one of the three colors: Red, Yellow, or Green. This again gives him 3 possible choices.
So, the total number of possible outcomes for his color assignments in both sessions is:
\[ 3 \text{ (first session)} \times 3 \text{ (second session)} = 9 \text{ total outcomes.} \]
Next, we identify the favorable outcomes where Julio is assigned the same color in both sessions. The favorable outcomes are:
- Red in both sessions.
- Yellow in both sessions.
- Green in both sessions.
Thus, there are 3 favorable outcomes (one for each color).
Now, we can calculate the probability that Julio will be assigned the same color for both sessions:
\[ P(\text{same color}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{9} = \frac{1}{3}. \]
Therefore, the probability that Julio will be assigned the same color for both sessions is
\[ \boxed{\frac{1}{3}}. \]