Question

Consider a political discussion group consisting of 9 ​Democrats, 8 ​Republicans, and 4 Independents. Suppose that two group members are randomly​ selected, in​ succession, to attend a political convention. Find the probability of selecting two Democrats.
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Part 1
  
enter your response here ​ (Type an integer or a simplified​ fraction.)

Answer 1

To find the probability of selecting two Democrats from the group, we'll follow these steps:

1. Calculate the total number of group members.
2. Find the probability of selecting a Democrat on the first and second selection.

### Step 1: Calculate the total number of group members
- Number of Democrats = 9
- Number of Republicans = 8
- Number of Independents = 4

Total members = 9 + 8 + 4 = 21

### Step 2: Find the probabilities

**Probability of selecting the first Democrat:**
- There are 9 Democrats out of 21 total members.

\[
P(\text{first Democrat}) = \frac{9}{21}
\]

**Probability of selecting the second Democrat:**
- If the first member selected is a Democrat, there will now be 8 Democrats left and 20 total members remaining.

\[
P(\text{second Democrat | first Democrat}) = \frac{8}{20}
\]

### Step 3: Calculate the combined probability
The probability of both events occurring (selecting two Democrats in a row) is the product of the two probabilities:

\[
P(\text{two Democrats}) = P(\text{first Democrat}) \times P(\text{second Democrat | first Democrat}) = \frac{9}{21} \times \frac{8}{20}
\]

Now simplify the fractions:

\[
\frac{9}{21} = \frac{3}{7} \quad (\text{dividing numerator and denominator by 3})
\]

\[
\frac{8}{20} = \frac{2}{5} \quad (\text{dividing numerator and denominator by 4})
\]

So now we have:

\[
P(\text{two Democrats}) = \frac{3}{7} \times \frac{2}{5} = \frac{6}{35}
\]

Thus, the probability of selecting two Democrats is:

\[
\frac{6}{35}
\]