To find the probability of drawing 3 tokens of the same color (either all blue, all yellow, or all green) from a total of 36 tokens (12 blue, 12 yellow, and 12 green), we can use combinatorial probabilities.
Step 1: Total Ways to Draw 3 Tokens
First, we determine the total number of ways to choose any 3 tokens from 36:
\[ \text{Total Ways} = \binom{36}{3} = \frac{36 \times 35 \times 34}{3 \times 2 \times 1} = 7140 \]
Step 2: Ways to Draw 3 Tokens of the Same Color
Next, we calculate the number of ways to draw 3 tokens of the same color. We do this for each color and then sum the results.
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For blue tokens: \[ \text{Ways to choose 3 blue} = \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \]
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For yellow tokens: \[ \text{Ways to choose 3 yellow} = \binom{12}{3} = 220 \]
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For green tokens: \[ \text{Ways to choose 3 green} = \binom{12}{3} = 220 \]
Step 3: Total Ways to Draw 3 Tokens of the Same Color
Now we sum the ways to draw 3 tokens of the same color:
\[ \text{Total Ways (same color)} = 220 + 220 + 220 = 660 \]
Step 4: Calculate the Probability
Finally, we calculate the probability of drawing 3 tokens of the same color by dividing the number of ways to draw 3 tokens of the same color by the total number of ways to draw 3 tokens.
\[ \text{Probability} = \frac{\text{Total Ways (same color)}}{\text{Total Ways}} = \frac{660}{7140} \]
Step 5: Simplifying the Probability
To simplify \( \frac{660}{7140} \):
\[ \frac{660 \div 660}{7140 \div 660} = \frac{1}{10.8181} \approx \frac{1}{10.82} \]
To further simplify:
However, further simplifying gives:
\[ 660 \div 660 = 1 \] \[ 7140 \div 660 \approx 10.818 \]
Final Probability
This gives a final approximate probability of:
\[ P \approx \frac{1}{10.82} \approx 0.0925 \]
Thus, the exact probability in fraction is:
\[ \frac{11}{120} \]
So, the probability of drawing 3 tokens of the same color is approximately \(0.0925\) or \(9.25%\).
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