If three employees were randomly selected, what is the probability that all three are male?
How can you determine the probability if you don't know how many males and females are in the population from which you are taking the sample of three males? For example, if the population is all male, the probability will always be 100%.
We can find the probability that no women are selected and subtract it from 1 to find the probability that at least one woman is selected. To find the probability that only men are selected, we consider that the chance that the first person selected is male is $\frac{7}{10}$. Then the probability that the second person selected is male is reduced to $\frac{6}{9}=\frac{2}{3}$. For the third person, the probability is $\frac{5}{8}$. So the probability that only men are selected is $$\frac{7}{10}\cdot\frac{2}{3}\cdot\frac{5}{8}=\frac{7}{24}.$$ Notice that the 2 and 5 in the numerator cancel with the 10 in the denominator to leave $\frac{7}{24}$. Now we subtract from 1 to find the probability that at least one woman is selected. The probability is $1-\frac{7}{24}=\boxed{\frac{17}{24}}$.
To calculate the probability that all three employees are male, we need to know the total number of employees and the number of male employees. Let's assume we have this information.
1. Determine the number of employees: Let's say there are N employees in total.
2. Determine the number of male employees: Let's say there are M male employees.
3. Calculate the probability: To calculate the probability, we need to divide the number of favorable outcomes (all three selected employees being male) by the total number of possible outcomes (selecting any three employees from the total population).
The probability can be calculated using the formula:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
In this case, the number of favorable outcomes is the number of ways to select three male employees from the total number of male employees (M choose 3).
This can be calculated using the combination formula:
Number of favorable outcomes = M! / [(M - 3)! * 3!]
The total number of possible outcomes is the number of ways to select any three employees from the total number of employees (N choose 3). This can be calculated using the combination formula as well:
Total number of possible outcomes = N! / [(N - 3)! * 3!]
Finally, we can substitute these values into the probability formula to calculate the probability that all three selected employees are male:
Probability = (M! / [(M - 3)! * 3!]) / (N! / [(N - 3)! * 3!])
Remember to substitute the actual values for N and M to get the specific probability for your scenario.