Final answer:
To find the union of two sets on a Venn diagram, draw the diagram with labeled circles for each set, shade the union area, and calculate combined probabilities, ensuring the total represents the entire sample space.
Step-by-step explanation:
Finding the union of two sets in a Venn diagram from word problems involves several steps. Here's a general approach:
- Draw a Venn diagram representing the situation. In a typical Venn diagram, you have a rectangle representing the universal set and circles within for each set in question.
- Label the sets according to the word problem. For example, if one set is novels (N) and the other nonfiction books (NF), you would draw two overlapping circles and label one N and the other NF.
- To represent the union of the sets (N U NF), you would shade in both circles entirely, indicating all elements that are either in set N, set NF, or in both sets.
- Describe the overlapping area in a complete sentence, such as "The overlapping area represents customers who buy both novels and nonfiction books."
- If you need to account for another set, such as customers who buy only compact disks (CD), draw another circle that does not overlap with the others and label it CD.
- To find the probability of a combined event (e.g., buying a novel or nonfiction book), add up the probabilities or frequencies within the union of the sets. Subtract any overlap if it's been counted twice. Make sure the total probabilities add up to 1 if it represents the entire sample space.
To solve for specific probabilities:
- Calculate the probability that a student belongs to a club (C).
- Compute the probability that a student works part-time (PT).
- Determine the probability for a student who belongs to both a club and works part-time (C AND PT).
- Find the conditional probability that a student belongs to a club given that they work part-time.
- Finally, calculate the probability that a student belongs to a club or works part-time (C OR PT).