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?
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If you randomly assign the physicians to the patients, you are randomly drawing from the latter number of possibilities. Each of these possibilities is equally likely, so the probability of any given possibility is 1 divided the total number ways are there to choose 4 out of the total of 9
So, if there are N possible ways to assign 4 female physicians to the patients, then the probability is N times the probability of one particular outcome.
Therefore the probability is the ratio of the number of ways you can choose 4 physicians out of the group of 5 females to the number of ways there are to choose 4 out of the total of 9.
To solve this problem, we need to find the probability that all the assigned physicians are female.
First, let's determine the total number of ways the 9 physicians can be assigned to the 4 patients. To do this, we will use combinations. The formula for combinations is nCr, where n is the total number of items and r is the number of items being chosen.
In this case, we have 9 physicians to choose from and we are assigning them to 4 patients. So, the number of ways they can be assigned is:
9C4 = (9!)/[(4!)(9-4)!] = 9!/(4!)(5!) = 9x8x7x6/(4x3x2x1) = 126
Next, let's determine the number of ways all the assigned physicians can be female. We have 5 female physicians to choose from, and we need to assign all of them to the 4 patients. So, the number of ways they can be assigned is:
5C4 = (5!)/[(4!)(5-4)!] = 5!/(4!)(1!) = 5x4x3x2x1/(4x3x2x1) = 5
Now, we can calculate the probability by dividing the number of ways all the assigned physicians can be female by the total number of ways the physicians can be assigned:
Probability = Number of ways all assigned physicians are female / Total number of ways to assign physicians
= 5/126 ≈ 0.0397
Therefore, the probability that all the assigned physicians are female is approximately 0.0397 or 3.97%.