Question

In a prize drawing, the tokens that participants will randomly select are numbered from 1 to 36 . 12 of
these tokens are blue,12 are yellow, and 12 are green. The first prize goes to the participant who
draws of the same color in a row without replacing tokens after each draw. What is the
probability of drawing3 tokens of the same color?

Answer 1

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.

- For blue tokens:
\[
\text{Ways to choose 3 blue} = \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\]

- For yellow tokens:
\[
\text{Ways to choose 3 yellow} = \binom{12}{3} = 220
\]

- 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 \(9.25\%\).