Final answer:
The probability of randomly selecting a budget committee with 4 Democrats and 4 Republicans from a pool of 18 Democrats and 7 Republicans is 0.035.
Step-by-step explanation:
The task is to determine the probability that a randomly formed committee of 8 congressmen will consist of 4 democrats and 4 republicans, given that there are 18 democrats and 7 republicans available. To calculate this, we can use the hypergeometric distribution, which is appropriate for scenarios where selections are made without replacement from two distinct groups.
First, we calculate the number of ways to choose 4 democrats out of 18, which is a combination denoted as C(18, 4). Likewise, we calculate the number of ways to choose 4 Republicans out of 7, denoted as C(7, 4). These two figures represent the favorable outcomes.
Next, we calculate the total number of ways to form a committee of 8 out of the total 25 congressmen, which is C(25, 8).
To find the probability, we divide the product of the favorable outcomes by the total number of outcomes:
P(4 democrats and 4 republicans) = (C(18, 4) * C(7, 4)) / C(25, 8)
Calculating the combinations:
- C(18, 4) = 18! / (4!(18-4)!) = 3060
- C(7, 4) = 7! / (4!(7-4)!) = 35
- C(25, 8) = 25! / (8!(25-8)!) = 1081575
So, the probability is:
P(4 democrats and 4 republicans) = (3060 * 35) / 1081575
After carrying out the calculations:
P(4 democrats and 4 republicans) = 0.035