In a freshman high school class of 80 students, 22 students take Consumer Education, 20 students take French, and 4' students take both. Which equation can be used to find the probability, P, that a randomly selected student from this class takes Consumer Education, French, or both?P= 11/40 + 1/4 + 1/20P= 11/40 + 1/4 - 1/20P= 11/40 + 1/4 - 1/10P= 11/40 + 1/4

• The probability of a student taking Consumer Education (\(P( \text { CE} ) \)) is \(\frac { 22} { 80} = \frac { 11} { 40} \).
• The probability of a student taking French (\(P( \text { F} ) \)) is \(\frac { 20} { 80} = \frac { 1} { 4} \).
• The probability of a student taking both (\(P( \text { CE} \cap \text { F} ) \)) is \(\frac { 4} { 80} = \frac { 1} { 20} \).
  Using the principle of inclusion-exclusion for probabilities:
  \[P( \text { CE} \cup \text { F} ) = P( \text { CE} ) + P( \text { F} ) - P( \text { CE} \cap \text { F} ) \]
  Substitute the values:
  \[P = \frac { 11} { 40} + \frac { 1} { 4} - \frac { 1} { 20} \]

Supplemental Knowledge

In probability, when dealing with events that can overlap (such as students taking both Consumer Education and French), we use the principle of inclusion-exclusion to avoid double-counting those who are part of both groups.

1. Principle of Inclusion-Exclusion:
   • To find the probability that a student takes either Consumer Education, French, or both, we need to add the probabilities of each individual event and then subtract the probability of the intersection (students taking both courses).
2. Calculating Probabilities:
   • The probability \(P( A) \) of an event \(A\) occurring is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.
   • For example, if 22 out of 80 students take Consumer Education, the probability is \(22/80\).

Concepts to Actions

Understanding combined probabilities is useful in many real-life circumstances, from event planning and marketing campaigns, where understanding which categories people might be most interested in can help better allocate resources; to school settings where it allows teachers and principals to schedule classes according to student interests or determine class attendance more easily.

Mastering concepts like probability and set theory can significantly enhance your analytical skills. At UpStudy, we provide comprehensive tools and resources to help you grasp these essential mathematical principles. Our live tutor question bank and AI-powered problem-solving services are designed to support your learning journey. 

To further assist with your mathematical calculations related to probabilities and other algebraic concepts, consider using UpStudy's Statistics and Probability Calculator. This tool can help you quickly solve complex problems and deepen your understanding of statistical principles!