Problem 24 - 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(2 sophomores)

Step-by-Step Solution

Answer

P(2 sophomores) = \( \frac{9}{20} \)

Step 1: Understand the Problem

We need to calculate the probability of selecting 2 sophomores when three students are chosen randomly from a group of 3 sophomores and 3 juniors. The table provides the probabilities for selecting 0, 1, 2, or 3 sophomores.

Step 2: Locate the Probability

Look at the probability distribution table provided: \[\begin{array}{|c|c|c|c|}\hline 0 & {1} & {2} & {3} \ \hline 1 & {\frac{9}{20}} & {\frac{9}{20}} & {\frac{1}{20}} \ \hline\end{array}\] Identify the probability associated with 2 sophomores, which is \( \frac{9}{20} \).

Key Concepts

Random Selection

Random selection occurs when each member of a group has an equal chance of being chosen. It’s important for ensuring fairness and eliminating bias in outcomes. In the context of probability, random selection allows us to focus on just the numbers and likelihoods without any external influence.
When choosing students from a group of 3 sophomores and 3 juniors, each student has an equal chance of being selected, regardless of their grade level. The probability of randomly selecting any particular combination, like 2 sophomores and 1 junior, will rely purely on the combinations possible and their respective likelihoods.

Probability Distribution

Probability distribution is a function or a table that provides the probabilities of various possible outcomes in an experiment. For any discrete random variable, like the number of sophomores chosen, the sum of all probabilities must equal 1.
In the original exercise, the table shows the probability distribution of choosing 0, 1, 2, or 3 sophomores out of the group of 3 sophomores and 3 juniors. This distribution tells us how likely each scenario is. For instance, the probability of selecting exactly 2 sophomores is given as \( \frac{9}{20} \).
Understanding this distribution allows students to quickly identify the likelihood of specific outcomes by referring to the given probabilities, like spotting the probability for 2 sophomores in the problem.

Combinatorics

Combinatorics is a branch of mathematics concerning the counting, arrangement, and combination of objects. It’s essential when dealing with situations involving random selection, where you want to know the number of possible ways to arrange or select objects.
In the problem, combinatorics helps us determine the number of ways to select 3 students from a mix of 3 sophomores and 3 juniors. Specifically, there are combinations like selecting 2 sophomores and 1 junior that we’re interested in.
To determine these possibilities, we often use combinations formulas, denoted as \( C(n, k) \), where \( n \) is the total number of items to choose from and \( k \) is the number of items to choose. This concept is key to understanding how the probability distributions are calculated and why certain outcomes are as likely as they are.