Textbook Question
1. When two events are mutually exclusive, why is P(A and B) = 0?
4. Probability
Addition Rule
2:47 minutesProblem 3.3.13
Textbook Question
13. Students A physics class has 40 students. Of these, 12 students are physics majors and 16 students are minoring in math. Of the physics majors, three are minoring in math. Find the probability that a randomly selected student is minoring in math or a physics major.
Verified step by step guidance1
Step 1: Identify the relevant sets and their sizes. The total number of students is 40. The number of physics majors is 12, the number of students minoring in math is 16, and the number of students who are both physics majors and minoring in math is 3.
Step 2: Use the formula for the union of two sets to calculate the probability. The formula is: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \), where \( A \) is the event of being a physics major and \( B \) is the event of minoring in math.
Step 3: Calculate \( P(A) \), the probability of being a physics major. This is given by \( \frac{\text{Number of physics majors}}{\text{Total number of students}} \), which is \( \frac{12}{40} \).
Step 4: Calculate \( P(B) \), the probability of minoring in math. This is given by \( \frac{\text{Number of students minoring in math}}{\text{Total number of students}} \), which is \( \frac{16}{40} \).
Step 5: Calculate \( P(A \cap B) \), the probability of being both a physics major and minoring in math. This is given by \( \frac{\text{Number of students who are both}}{\text{Total number of students}} \), which is \( \frac{3}{40} \). Substitute these values into the union formula to find \( P(A \cup B) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the chance of selecting a student who is either a physics major or minoring in math from a total group of students. Understanding how to compute probabilities is essential for solving the given question.
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Union of Events
The union of two events refers to the occurrence of at least one of the events. In this case, we are interested in the union of the events 'being a physics major' and 'minoring in math.' The formula for the probability of the union of two events is P(A ∪ B) = P(A) + P(B) - P(A ∩ B), which accounts for any overlap between the two groups.
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Set Theory
Set theory provides a framework for understanding collections of objects, which in this case are the students categorized by their academic focus. By using set notation, we can define the groups of physics majors and math minors, and analyze their intersections and unions. This helps in visualizing and calculating the probabilities of the combined events effectively.
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