Math Problem Statement

A marble is selected from a bag containing eight marbles numbered 1 to 8. The number on the marble selected will be recorded as the outcome.

Consider the following events.

Event A: The marble selected has a number less than 4. Event B : The marble selected has an odd number. Give the outcomes for each of the following events. If there is more than one element in the set, separate them with commas.

Solution

Let's start by identifying the numbers that satisfy each event:

  • Event A: The marble selected has a number less than 4.

    • The numbers that are less than 4 are: 1,2,31, 2, 3.
    • Therefore, the outcomes for Event A are: {1,2,3}\{1, 2, 3\}.
  • Event B: The marble selected has an odd number.

    • The odd numbers between 1 and 8 are: 1,3,5,71, 3, 5, 7.
    • Therefore, the outcomes for Event B are: {1,3,5,7}\{1, 3, 5, 7\}.

Now, let's define any possible combined events:

  1. ABA \cap B (Event A AND Event B): The marble selected has a number less than 4 and is odd.

    • The common numbers in both Event A and Event B are: 1,31, 3.
    • So, AB={1,3}A \cap B = \{1, 3\}.
  2. ABA \cup B (Event A OR Event B): The marble selected has a number less than 4 or is odd.

    • This includes all the numbers in either Event A or Event B: 1,2,3,5,71, 2, 3, 5, 7.
    • So, AB={1,2,3,5,7}A \cup B = \{1, 2, 3, 5, 7\}.
  3. AcA^c (The complement of Event A): The marble selected has a number greater than or equal to 4.

    • The numbers greater than or equal to 4 are: 4,5,6,7,84, 5, 6, 7, 8.
    • So, Ac={4,5,6,7,8}A^c = \{4, 5, 6, 7, 8\}.

Would you like further clarification or details on any of the events?

Here are 5 related questions to expand on the concept:

  1. What is the probability of Event A happening?
  2. What is the probability of Event B happening?
  3. What is the probability of ABA \cap B occurring?
  4. What is the probability of ABA \cup B occurring?
  5. How would the events change if the bag had marbles numbered 1 to 10?

Tip: To find the probability of an event, divide the number of favorable outcomes by the total number of outcomes.

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Math Problem Analysis

Mathematical Concepts

Probability
Set Theory

Formulas

A ∩ B = A AND B
A ∪ B = A OR B
A^c = complement of A

Theorems

Intersection of sets
Union of sets
Complement of a set

Suitable Grade Level

Grades 6-8