Problem 23 - 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(1 sophomore)

Step-by-Step Solution

Answer

The probability of picking 1 sophomore is \( \frac{9}{20} \).

Step 1: Understand the Distribution Table

The table shows the distribution of the number of sophomores chosen when picking three students. The top row represents the number of sophomores chosen (0, 1, 2, or 3), and the second row gives the relative frequency (probability) for each outcome.

Step 2: Identify the Required Probability

We are asked to find the probability of choosing 1 sophomore. In the distribution table, locate the column corresponding to '1' in the first row.

Step 3: Read the Probability Value From the Table

In the identified column, look for the relative frequency beneath '1' in the table. The frequency corresponding to choosing 1 sophomore is given as \( \frac{9}{20} \).

Step 4: Confirm the Probability as a Value

The relative frequency directly provides the probability of the event occurring. Therefore, the probability \( P(1 \text{ sophomore}) \) is equal to \( \frac{9}{20} \).

Key Concepts

Relative-Frequency Histogram

A relative-frequency histogram is a graphical way of displaying how often different outcomes occur relative to each other. Imagine you have data showcasing the outcomes of an event, like the one involving sophomores and juniors in this exercise. The height of each bar in a relative-frequency histogram represents the proportion of the total count.

In our context, the histogram helps visualize the probabilities of selecting 0, 1, 2, or 3 sophomores when choosing three students. This way, you can easily see which outcome is most common and which is least expected. Relative frequencies range from 0 to 1, with each bar illustrating how likely or unlikely it is to select a particular number of sophomores.

- Visualizes probabilities - Heights represent frequency out of total - Easy comparison across different outcomes

Distribution Table

A distribution table presents data points and their probabilities in a structured way. For our problem, it reveals how many sophomores you might draw from a mixture of sophomores and juniors, alongside the probability of each occurrence.

You'll notice two rows here. The top row details possible outcomes (how many sophomores), while the second row indicates the probability (or relative frequency) associated with each occurrence. Understanding this setup helps in quickly identifying probabilities. For instance, if you want to find the chance of selecting exactly one sophomore, you'd locate '1' in the top row and then look beneath it to see the probability.

- Outputs covered are 0 to 3 sophomores - Possibly simplified selection probability
- Quick retrieval of probability information

Random Selection

Random selection is an unpredictable process where each item in a group has an equal opportunity of being chosen. In this exercise, this concept applies to choosing students from a mix of sophomores and juniors.

When we decide to select students randomly, no bias is introduced. Each student, whether sophomore or junior, has the same chance of being part of our selection. This impartiality is crucial to ensure fairness in experimental or survey conditions. Here, probability calculations rely on such randomness, reflecting the theoretical expectations based on equally likely events.

- Equal chance of selection for all - Ensures unbiased results
- Foundation for calculating true probabilities

Sophomores and Juniors

In this probability scenario, we are dealing with two specific groups within a body of students: sophomores and juniors. Each class year represents a different set within our selection process, adding diversity to potential outcomes.

Understanding these groups helps clarify what's meant by selecting 'x' number of sophomores, as it directly affects how the outcome probabilities are distributed. You have an equal number of sophomores and juniors, making it interesting to analyze how likely you are to pick more from one particular class.

Remember, the concept here helps us understand the scenario better—a key step to solve the probability distribution effectively.

- Sophomore and junior balance insights
- Diversity in possible outcomes
- Influence on probability spread