A college faculty consists of 400 men and 250 women. The college administration wants to draw a sample of 65 faculty members to ask their opinion about a new parking fee. They draw a simple random sample of 40 men and another simple random sample of 25 women.
To calculate the probability of the sample, we need to find out the probability of selecting a specific number of men and women from the faculty.
Let's start with the probability of selecting 40 men from the faculty of 400 men.
The probability of selecting one man at random is 400/650 (since you have 400 men and 650 total faculty members).
To find the probability of selecting 40 men, we need to use the formula for combinations, which is denoted as nCr:
nCr = n! / (r! * (n-r)!)
where n is the total number of items, r is the number of items chosen, and ! represents the factorial of a number.
So, applying this formula:
40 men from 400 men: 400C40 = 400!/ (40! * (400-40)!)
Similarly, to calculate the probability of selecting 25 women from the faculty of 250 women:
25 women from 250 women: 250C25 = 250! / (25! * (250-25)!)
Now, we can calculate the probability of selecting the sample of 65 faculty members. Since we need to select 40 men and 25 women, we multiply the probabilities of selecting these specific groups:
Probability of selecting the sample = (400C40 * 250C25) / total possible combinations
The total possible combinations can be calculated by using nCr, where n is the total number of faculty members (650 in this case) and r is the sample size (65 in this case):
Total possible combinations = 650C65 = 650! / (65! * (650-65)!)
With these calculations, you can determine the probability of selecting the given sample using simple random sampling.