A clinic employs 9 physicians. 5 of the physicians are female. 4 patients arrive at once. Assuming the doctors are assigned randomly to patients, what is the probability that all of the assigned physicians are female?
.0397
5/9 x 4/8 x 3/7 x 2/6 = ?
To calculate the probability that all assigned physicians are female, we need to determine the number of ways that all the physicians can be assigned and the number of ways that only the female physicians can be assigned.
First, let's find the number of ways that all 9 physicians can be assigned to the 4 patients. Since the assignment is random, we can use the concept of combinations. The formula for combinations is:
nCr = n! / (r!(n-r)!)
where n is the total number of items and r is the number of items chosen.
In this case, we want to find the number of ways to choose 4 physicians out of the 9 available. So the number of ways that all the physicians can be assigned is:
9C4 = 9! / (4!(9-4)!)
= 9! / (4!5!)
= (9*8*7*6) / (4*3*2*1)
= 126
Now, let's find the number of ways that only the 5 female physicians can be assigned. We want to choose 4 out of the 5 female physicians. So the number of ways that only the female physicians can be assigned is:
5C4 = 5! / (4!(5-4)!)
= 5! / (4!1!)
= 5
Therefore, the probability that all the assigned physicians are female is:
P(all assigned physicians are female) = (number of ways to assign only female physicians) / (total number of ways to assign all physicians)
= 5 / 126
≈ 0.0397 or 3.97% (rounded to 2 decimal places)
So the probability that all of the assigned physicians are female is approximately 0.0397 or 3.97%.
To find the probability that all of the assigned physicians are female, we need to consider the number of ways to assign the physicians to patients and the number of favorable outcomes where all of the assigned physicians are female.
First, let's calculate the total number of ways to assign the physicians. Since there are 9 physicians and 4 patients, we have 9 choices for the first patient, 8 choices for the second patient, 7 choices for the third patient, and 6 choices for the fourth patient. Therefore, the total number of ways to assign the physicians is 9 * 8 * 7 * 6.
Next, let's calculate the number of favorable outcomes where all of the assigned physicians are female. Since there are 5 female physicians out of the 9 total physicians, we have 5 choices for the first patient, 4 choices for the second patient, 3 choices for the third patient, and 2 choices for the fourth patient. Therefore, the number of favorable outcomes is 5 * 4 * 3 * 2.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
P(all physicians are female) = (number of favorable outcomes) / (total number of outcomes)
P(all physicians are female) = (5 * 4 * 3 * 2) / (9 * 8 * 7 * 6)
Simplifying the equation:
P(all physicians are female) = 0.0952
Therefore, the probability that all of the assigned physicians are female is approximately 0.0952 or 9.52%.