Problem 22 - 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(0 sophomores)

Step-by-Step Solution

Answer

The probability is 1.

Step 1: Understanding the Problem

We need to find the probability of selecting 0 sophomores when 3 students are chosen at random from a group of 3 sophomores and 3 juniors. The probability is given in a table: the value corresponds to choosing 0 sophomores.

Step 2: Locate the Probability in the Table

The table shows probabilities for choosing 0, 1, 2, or 3 sophomores. Look at the first position in the table under the heading for 0 sophomores. The probability shown is what we need to extract.

Step 3: Extract Probability Value

According to the table, the probability of selecting 0 sophomores is given as \(1\). Thus, P(0 sophomores) = \(1\).

Key Concepts

Relative-Frequency Histogram

A relative-frequency histogram is a visual tool that helps us understand the distribution of data. It shows the frequency of different outcomes in relation to each other. In probability, this type of histogram demonstrates the relative likelihood of various events occurring based on experimental or observational data.

When dealing with probabilities, the total area of a relative-frequency histogram equals 1. Each bar's height represents the probability of that particular event. In our example, we have a histogram that shows the distribution of chosen sophomores when selecting randomly from sophomores and juniors.

The histogram allows us to easily compare which outcomes are more likely. Here, bars correspond to selecting 0, 1, 2, or 3 sophomores. Using this tool, identifying probabilities such as selecting 0 sophomores becomes clearer and more intuitive.

Combinations

Combinations are a fundamental concept used to determine how many ways you can select items from a larger set. In probability, they help in counting the number of possible outcomes without regard to the order.

For instance, when choosing students from a group, it involves selecting a subset of students. To calculate combinations, we use the formula: \[ C(n, k) = \frac{n!}{k!(n-k)!} \] where \( n \) is the total number of items, and \( k \) is the number of items to select.

In the given problem, we have combinations of 3 sophomores from a total group of 6 students to figure out how many arrangements feature 0 sophomores. This simplifies the calculation by focusing only on the selection aspect rather than their arrangement.

Random Selection

Random selection is the process of choosing items or individuals in such a way that each has an equal chance of being selected. This concept is central to probability because it ensures fairness and eliminates bias when making selections.

In the context of the exercise, random selection occurs when choosing 3 students out of a total of 6 (3 sophomores and 3 juniors). Each student has an equal likelihood of being chosen.

The random selection ensures that each outcome is based purely on chance, helping when calculating probabilities. This forms the basis for assuming that each of the different combinations of selections is equally probable.