Problem 51 - Algebra and Trigonometry

Question

Choosing a Committee A committee of five is chosen randomly from a group of six males and eight females. What is the probability that the committee includes either all males or all females?

Step-by-Step Solution

Answer

The probability of choosing a committee with either all males or all females is approximately 0.031.

Step 1: Define Total Possible Committees

First, we need to find the total number of ways to choose a committee of 5 people from the entire group of 14 people (6 males + 8 females). This can be calculated using the combination formula: \( \binom{n}{k} \), where \( n \) is the total number of people to choose from, and \( k \) is the number of people to choose. Here, \( n = 14 \) and \( k = 5 \), so the total number of ways is \( \binom{14}{5} \).

Step 2: Calculate Total Possible Committees

Calculate the combination \( \binom{14}{5} \): \[ \binom{14}{5} = \frac{14 \times 13 \times 12 \times 11 \times 10}{5 \times 4 \times 3 \times 2 \times 1} = 2002 \]So, there are 2002 possible committees.

Step 3: Number of All Males Committee

Now calculate the number of ways to choose 5 males. Since there are only 6 males, the combination is \( \binom{6}{5} \).

Step 4: Calculate All Males Committee

Calculate the combination \( \binom{6}{5} \): \[ \binom{6}{5} = \frac{6}{1} = 6 \]So, there are 6 ways to make a committee of 5 males.

Step 5: Number of All Females Committee

Now calculate the number of ways to choose 5 females. Since there are 8 females, the combination is \( \binom{8}{5} \).

Step 6: Calculate All Females Committee

Calculate the combination \( \binom{8}{5} \): \[ \binom{8}{5} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]So, there are 56 ways to make a committee of 5 females.

Step 7: Calculate Total Specific Committees

Add the number of ways to form the all-male committee and the all-female committee. Thus, it is \( 6 + 56 = 62 \).

Step 8: Calculate Probability

The probability that the committee is either all males or all females is the number of favorable outcomes (62) divided by the total number of possible outcomes (2002): \[ P = \frac{62}{2002} \approx 0.030969 \] which is approximately 0.031.

Key Concepts

Combinations

When selecting groups or subsets without regard to the order in which the elements are chosen, we use a concept called combinations. In mathematics, these are often represented by the term "binomial coefficient."
This is typically expressed using notation like \( \binom{n}{k} \), which represents the number of ways to choose \( k \) elements from a set of \( n \) elements.

• For example, choosing 5 people out of a group of 14 uses combinations to determine how many unique committees you could form.
• The formula is \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).

This formula allows us to focus on the selection itself rather than any kind of arrangement or sequence of the chosen members.
Essentially, it's about picking members for a group, and here the order of selection doesn't impact the outcome.

Committee Selection

In the context of probability and combinations, committee selection refers to the process of forming a specific group from a larger pool.
This can involve selecting members for a particular purpose, such as assigning participants into a committee.

• For instance, to form a committee of 5 members from a total of 14 people, you calculate using combinations as shown above.
• For specific scenarios, like forming a committee with all members from the same gender, you calculate the possible all-male and all-female committees separately.

This ensures that you understand the different ways a committee can be configured based on given criteria, thereby framing the probability scenario clearly. This approach simplifies the complex setup of choosing groups for targeted characteristics like all-male or all-female combinations.

Binomial Coefficient

The binomial coefficient is a key mathematical concept used in the calculation of combinations. It provides a way to determine the number of ways to choose \( k \) elements from a set without regard to the order.
In scenarios like committee formations, binomial coefficients make it easier to compute potential groupings rapidly.

• The binomial coefficient is represented by \( \binom{n}{k} \) and calculated using the formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
• This framework is essential in various applications, from problems in probability to functions in algebra.

In the exercise mentioned, calculating the total number of committees, or specific types like all-male or all-female committees,relies heavily on this coefficient.
Understanding how it operates helps grasp why it simplifies complex selection tasks efficiently.