Problem 22 - Algebra and Trigonometry

Question

The president of a large company selects six employees to receive a special bonus. He claims that the six employees are chosen randomly from among the 30 employees, of which 19 are women and 11 are men. What is the probability that no woman is chosen?

Step-by-Step Solution

Answer

The probability that no woman is chosen is approximately 0.0007782.

Step 1: Understanding the Problem

The problem asks us to find the probability that all six employees chosen for a bonus are men. We have 11 men and 19 women in total, making a total of 30 employees.

Step 2: Total Ways to Choose Employees

First, find the total number of ways to choose any 6 employees from the 30 employees. This is a combination problem, and the formula for combinations is \( \binom{n}{k} \), where \( n \) is the total number and \( k \) is the number chosen. Thus, the total number of ways is \( \binom{30}{6} \).

Step 3: Ways to Choose Only Men

Next, determine the number of ways to choose 6 men from the 11 available. Again, this is a combination problem: \( \binom{11}{6} \) ways.

Step 4: Calculating the Probability

The probability that all 6 selected are men is given by the fraction of favorable outcomes (all men) divided by the total outcomes (any 6 employees). Therefore, the probability is \( \frac{\binom{11}{6}}{\binom{30}{6}} \). Calculate this probability to find the answer.

Step 5: Performing the Calculations

Calculate \( \binom{11}{6} \) which is the number of ways to choose 6 men from 11. Then calculate \( \binom{30}{6} \), the total number of ways to choose any 6 employees. Using these calculations, find the probability: \[ \frac{\binom{11}{6}}{\binom{30}{6}} = \frac{462}{593775} \approx 0.0007782. \]

Key Concepts

Understanding Combinations in Probability

Combinations are an essential concept in probability and statistics. They help us to determine how many ways we can select items from a larger set when the order does not matter. This is particularly useful in probability problems, where we often need to know how many ways a specific event can occur.
To calculate combinations, we use the notation \( \binom{n}{k} \). Here, \( n \) represents the total number of items, while \( k \) represents the number of items we want to select.

• For example, in the problem above, the total items are 30 employees, and the president selects 6 employees. Thus, \( \binom{30}{6} \) gives us the total number of combinations.
• If we specifically wanted to choose only men, we would look at the 11 men available and decide to pick 6 of them, which is \( \binom{11}{6} \).

The combination formula is \( \binom{n}{k} = \frac{n!}{k! (n-k)!} \). This formula helps us calculate the number of ways items can be selected without considering the order.

Analyzing Total Outcomes vs Favorable Outcomes

Probability often involves distinguishing between the total number of possible outcomes and the specific number of favorable outcomes you are interested in. Understanding this difference is key to solving probability problems correctly.
In the example, the total number of outcomes is based on the total ways we can choose any 6 employees from the group of 30. This is why we calculate \( \binom{30}{6} \).

• Total Outcomes: The entire set of possibilities, e.g., all possible groups of 6 employees chosen from 30.
• Favorable Outcomes: The subsets of the total outcomes that fit our criteria, e.g., groups of 6 men chosen from 11.

By calculating \( \binom{11}{6} \), we find the total number of groups that include only men, which represents our favorable outcomes in this scenario.

Performing Probability Calculations

Once we understand combinations and distinguish between total and favorable outcomes, we proceed to calculate the probability itself. The formula for calculating probability is:
\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \]This formula gives us a numerical value representing how likely it is for the favorable outcomes to occur out of all possible outcomes.

• In the problem, we are calculating the probability that all chosen employees are men. We use the favorable outcome \( \binom{11}{6} \) and divide it by the total outcomes \( \binom{30}{6} \).
• The calculation results in: \[ \frac{462}{593775} \approx 0.0007782 \]

This probability tells us that the likelihood of all 6 selected employees being men is quite low, approximately 0.07782%.