Problem 25 - Algebra 2

Question

Three students are selected at random from a group of 3 sophomores and 3 juniors. The table and relative-frequency histogram show the distribution of the number of sophomores chosen. Find each probability. \(\begin{array}{|c|c|c|c|}\hline 0 & {1} & {2} & {3} \\ \hline 1 & {\frac{9}{20}} & {\frac{9}{20}} & {\frac{1}{20}} \\ \hline\end{array}\) P(3 sophomores)

Step-by-Step Solution

Answer

The probability of selecting 3 sophomores is \( \frac{1}{20} \).

Step 1: Understanding the Problem

The table and histogram provide the probabilities of selecting 0, 1, 2, or 3 sophomores when choosing 3 students randomly from a group consisting of 3 sophomores and 3 juniors. We need to find the probability that all selected students are sophomores.

Step 2: Extracting the Probability

The probability for each scenario (0, 1, 2, or 3 sophomores) is given directly in the table. We are interested in the scenario where all 3 students chosen are sophomores. From the table, this is the column labeled '3'. The probability assigned to this event is \( \frac{1}{20} \).

Step 3: Verifying the Probability Calculation

To be certain the table has been interpreted correctly, recalculate probabilities by considering combinations: There is only 1 way to choose all 3 students as sophomores out of 6 students. Calculate: \( \binom{3}{3} \times \binom{3}{0} / \binom{6}{3} = 1 \times 1 / 20 = \frac{1}{20} \).

Step 4: Confirming the Table Data Interpretation

Comparing the calculated value with the table value, both yield a probability of \( \frac{1}{20} \), confirming the table presents the correct data.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. When solving problems involving random selections, combinatorics helps determine the number of possible combinations.
For example, in the exercise above, the problem involved choosing 3 students from a group of 3 sophomores and 3 juniors. The total number of ways to choose 3 students from these 6 is calculated using combinatorics. This is often represented using binomial coefficients, which are a key component of combinatorics.
Understanding combinatorics is crucial for calculating probabilities when dealing with various configurations and selections. It simplifies your process in determining how certain events can occur or be arranged. Clear grasping of this concept is thus foundational in probability and statistics.

Relative-Frequency Histogram

A relative-frequency histogram is a graphical representation of data where the frequencies are portrayed as fractions of the whole, rather than pure counts. This type of histogram allows us to easily visualize probabilities.
In the exercise, the histogram corresponds with the table data, displaying the relative frequency of selecting a certain number of sophomores.

• The bar for 0 sophomores has a relative frequency.
• Similarly, bars for 1, 2, and 3 sophomores indicate their respective relative frequencies.

This visualization helps quickly understand how likely each different scenario is, by showing each probability visually. It can be extremely useful for analyzing the distribution of data visually rather than relying solely on numerical data.

Random Selection

Random selection is the process of choosing items from a group without any specific order or pattern. For probabilities to hold true, each member of the group must have an equal chance of being selected.
In the context of this exercise, random selection means choosing 3 students from 6 (3 sophomores and 3 juniors) with equal likelihood for each student.
This concept is fundamental in experiments and studies where fair representation is key. Random selection ensures no bias interferes with the choosing process, which is critical in deriving accurate probabilities and making valid statistical inferences.

Binomial Coefficient

A binomial coefficient is a mathematical symbol often denoted as \( \binom{n}{k} \), which represents the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order of selection. It's a cornerstone of combinatorics used to solve numerous probability problems.
Consider the problem scenario: we calculate the ways to choose 3 students all being sophomores from the group using \( \binom{3}{3} \). This results in one possible combination since all are sophomores.
To understand probabilities, mastering the use and calculation of binomial coefficients is critical. They simplify expressions where multiple scenarios can arise from a selection process, providing clarity and precision in finding solutions.