Question

Employees of a firm receive annual ratings. In a certain department, 4 employees received excellent ratings, 15 received good ratings, and 1 received marginal rating. If 3 employees in the department are randomly selected to complete a form for an internal study of the firm, find the probability that all 3 were rated excellent

Answer 1

Conditions

Employees of a firm receive annual ratings. In a certain department, 4 employees received excellent ratings, 15 received good ratings, and 1 received marginal rating. If 3 employees in the department are randomly selected to complete a form for an internal study of the firm, find the probability that all 3 were rated excellent

Solution

Let's use the Classic Probability Definition. It claims, that the probability of some event A is a ratio of a number of all favorable outcomes for this event to a number of all possible outcomes.

The number of all favorable outcomes is the amount of all possible ways, how to choose 3 employees from 4 with excellent ratings:

$$C _ {4} ^ {3} = \frac {4 !}{1 ! 3 !} = 4$$

The number of all possible outcomes is the amount of all possible ways, how to choose 3 employees from 4+15+1=20 employees:

$$C _ {2 0} ^ {3} = \frac {2 0 !}{1 7 ! 3 !} = \frac {1 8 - 1 9 - 2 0}{1 - 2 - 3} = 3 - 1 9 - 2 0 = 1 1 4 0$$ $$P = \frac {4}{1 1 4 0} = \frac {1}{2 8 5}$$

Answer: $\frac{1}{285}$