Lesson 47 Probabilities And Venn Diagrams Answers

Let's explore the fascinating world of probabilities using Venn diagrams, a powerful tool for visualizing and solving probability problems. Venn diagrams help us understand the relationships between different events and calculate probabilities accurately, especially when dealing with overlapping events.

Understanding Probabilities with Venn Diagrams

Venn diagrams are visual representations of sets and their relationships. In the context of probability, each set represents an event, and the overlapping areas represent the intersection of these events. The universal set represents all possible outcomes. By using Venn diagrams, we can clearly see the probabilities of different events occurring, both individually and in combination.

Key Concepts

Before diving into problem-solving, let's define some key probability concepts:

• Probability: The likelihood of an event occurring, expressed as a number between 0 and 1 (or as a percentage).
• Event: A specific outcome or set of outcomes.
• Sample Space: The set of all possible outcomes.
• Union (A ∪ B): The event where either A or B or both occur.
• Intersection (A ∩ B): The event where both A and B occur.
• Complement (A'): The event where A does not occur.
• Conditional Probability (P(A|B)): The probability of event A occurring given that event B has already occurred.
• Mutually Exclusive Events: Events that cannot occur at the same time (their intersection is empty).
• Independent Events: Events where the occurrence of one does not affect the probability of the other.

Basic Venn Diagram Setup

A basic Venn diagram consists of a rectangle representing the sample space and circles within the rectangle representing different events. The area within each circle represents the probability of that event occurring. Overlapping areas represent the probability of both events occurring.

Solving Probability Problems Using Venn Diagrams: Step-by-Step

Here's a structured approach to solving probability problems using Venn diagrams:

1. Read and Understand the Problem: Carefully analyze the problem statement to identify the events, probabilities, and relationships involved.

2. Draw the Venn Diagram:

   • Draw a rectangle to represent the sample space.
   • Draw circles within the rectangle to represent the events mentioned in the problem. The number of circles will depend on the number of events.
   • If the events overlap, make sure the circles intersect.

3. Fill in the Venn Diagram:

   • Start by filling in the intersection of the events (the overlapping areas). This is usually the most direct piece of information given.
   • Work outwards, using the given probabilities to deduce the probabilities of the remaining regions within each circle.
   • Remember that the sum of all probabilities within the Venn diagram must equal 1.

4. Calculate the Required Probabilities:

   • Use the filled-in Venn diagram to calculate the probabilities requested in the problem.
   • Apply the concepts of union, intersection, complement, and conditional probability as needed.

5. Check Your Answer:

   • Make sure the probabilities make sense within the context of the problem.
   • Double-check your calculations to avoid errors.

Example Problems and Solutions

Let's work through some example problems to illustrate the process.

Problem 1:

In a class of 30 students, 18 are taking mathematics, 10 are taking physics, and 5 are taking both mathematics and physics. What is the probability that a randomly selected student is taking either mathematics or physics?

Solution:

1. Understand the Problem:

   • Event A: Student is taking mathematics.
   • Event B: Student is taking physics.
   • Total number of students: 30
   • Number of students taking mathematics: 18
   • Number of students taking physics: 10
   • Number of students taking both: 5

2. Draw the Venn Diagram:

   • Draw a rectangle representing all 30 students.
   • Draw two overlapping circles, one for mathematics (A) and one for physics (B).

3. Fill in the Venn Diagram:

   • The intersection (A ∩ B) has 5 students.
   • The number of students taking only mathematics is 18 - 5 = 13.
   • The number of students taking only physics is 10 - 5 = 5.
   • The number of students taking neither is 30 - (13 + 5 + 5) = 7.

4. Calculate the Required Probabilities:

   • We want to find the probability that a student is taking either mathematics or physics, which is P(A ∪ B).
   • P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
   • P(A ∪ B) = (18/30) + (10/30) - (5/30) = 23/30

5. Check Your Answer:

   • The probability 23/30 makes sense; it's less than 1 and represents a significant portion of the class.

Answer: The probability that a randomly selected student is taking either mathematics or physics is 23/30.

Problem 2:

A survey was conducted among 100 people to find out their preference for tea and coffee. 60 people like tea, 50 people like coffee, and 20 people like both. What is the probability that a person selected at random likes neither tea nor coffee?

Solution:

1. Understand the Problem:

   • Event A: Person likes tea.
   • Event B: Person likes coffee.
   • Total number of people: 100
   • Number of people who like tea: 60
   • Number of people who like coffee: 50
   • Number of people who like both: 20

2. Draw the Venn Diagram:

   • Draw a rectangle representing all 100 people.
   • Draw two overlapping circles, one for tea (A) and one for coffee (B).

3. Fill in the Venn Diagram:

   • The intersection (A ∩ B) has 20 people.
   • The number of people who like only tea is 60 - 20 = 40.
   • The number of people who like only coffee is 50 - 20 = 30.
   • The number of people who like neither is 100 - (40 + 20 + 30) = 10.

4. Calculate the Required Probabilities:

   • We want to find the probability that a person likes neither tea nor coffee, which is P((A ∪ B)').
   • P((A ∪ B)') = Number of people who like neither / Total number of people
   • P((A ∪ B)') = 10/100 = 1/10

5. Check Your Answer:

   • The probability 1/10 makes sense; it's a small portion of the total population.

Answer: The probability that a person selected at random likes neither tea nor coffee is 1/10.

Problem 3:

In a group of 50 students, 30 are good at mathematics, 25 are good at English, and 10 are good at both. If a student is selected at random, find the probability that:

a) He is good at mathematics but not in English. b) He is good at English but not in mathematics. c) He is good at either mathematics or English.

Solution:

1. Understand the Problem:

   • Event A: Student is good at mathematics.
   • Event B: Student is good at English.
   • Total number of students: 50
   • Number of students good at mathematics: 30
   • Number of students good at English: 25
   • Number of students good at both: 10

2. Draw the Venn Diagram:

   • Draw a rectangle representing all 50 students.
   • Draw two overlapping circles, one for mathematics (A) and one for English (B).

3. Fill in the Venn Diagram:

   • The intersection (A ∩ B) has 10 students.
   • The number of students good at only mathematics is 30 - 10 = 20.
   • The number of students good at only English is 25 - 10 = 15.
   • The number of students good at neither is 50 - (20 + 10 + 15) = 5.

4. Calculate the Required Probabilities:

   • a) He is good at mathematics but not in English: This is P(A ∩ B') = 20/50 = 2/5
   • b) He is good at English but not in mathematics: This is P(B ∩ A') = 15/50 = 3/10
   • c) He is good at either mathematics or English: This is P(A ∪ B) = (20 + 10 + 15)/50 = 45/50 = 9/10

5. Check Your Answer:

   • All probabilities are between 0 and 1 and seem reasonable given the information.

Answers:

• a) 2/5
• b) 3/10
• c) 9/10

Problem 4:

In a survey of 200 people about their reading habits, it was found that 80 read magazines, 90 read novels, and 40 read both. Find the probability that a person chosen at random:

a) Reads magazines or novels. b) Reads neither magazines nor novels. c) Reads magazines if it is known that they read novels.

Solution:

1. Understand the Problem:

   • Event A: Person reads magazines.
   • Event B: Person reads novels.
   • Total number of people: 200
   • Number of people who read magazines: 80
   • Number of people who read novels: 90
   • Number of people who read both: 40

2. Draw the Venn Diagram:

   • Draw a rectangle representing all 200 people.
   • Draw two overlapping circles, one for magazines (A) and one for novels (B).

3. Fill in the Venn Diagram:

   • The intersection (A ∩ B) has 40 people.
   • The number of people who read only magazines is 80 - 40 = 40.
   • The number of people who read only novels is 90 - 40 = 50.
   • The number of people who read neither is 200 - (40 + 40 + 50) = 70.

4. Calculate the Required Probabilities:

   • a) Reads magazines or novels: P(A ∪ B) = (40 + 40 + 50)/200 = 130/200 = 13/20
   • b) Reads neither magazines nor novels: P((A ∪ B)') = 70/200 = 7/20
   • c) Reads magazines if it is known that they read novels: This is conditional probability P(A|B) = P(A ∩ B) / P(B) = (40/200) / (90/200) = 40/90 = 4/9

5. Check Your Answer:

   • All probabilities are between 0 and 1 and seem reasonable.

Answers:

• a) 13/20
• b) 7/20
• c) 4/9

Advanced Applications of Venn Diagrams in Probability

Beyond basic problems, Venn diagrams can be used to solve more complex probability scenarios involving multiple events and conditional probabilities. Here are some advanced applications:

Three-Event Venn Diagrams

When dealing with three events, the Venn diagram becomes more intricate, with multiple overlapping regions. The key is to systematically fill in the probabilities, starting with the intersection of all three events, and then working outwards to the intersections of two events and finally the individual events.

Conditional Probability with Venn Diagrams

Venn diagrams provide a visual way to understand conditional probability. P(A|B) represents the probability of event A occurring given that event B has already occurred. In the Venn diagram, this means we are only considering the portion of the diagram that represents event B, and then finding the proportion of that area that also includes event A (the intersection).

Independence of Events

Two events A and B are independent if the occurrence of one does not affect the probability of the other. Mathematically, this means P(A ∩ B) = P(A) * P(B). In a Venn diagram, independent events are not visually obvious, but you can verify independence by calculating the probabilities and checking if the equation holds true.

Common Mistakes to Avoid

• Forgetting the Universal Set: Always remember that the sum of all probabilities within the Venn diagram must equal 1. The area outside the circles is just as important as the areas inside.
• Incorrectly Filling in the Intersections: Start with the innermost intersection (the intersection of all events) and work outwards. Make sure you are subtracting the appropriate values when calculating the probabilities of the "only" regions.
• Misinterpreting Conditional Probability: Remember that P(A|B) is not the same as P(B|A). The order matters.
• Assuming Independence Without Verification: Don't assume that events are independent unless you are explicitly told so or you can verify it mathematically.

Tips for Success

• Practice Regularly: The more you practice, the more comfortable you will become with using Venn diagrams to solve probability problems.
• Draw Neat and Clear Diagrams: A well-drawn Venn diagram can make the problem much easier to understand and solve.
• Label Everything Clearly: Label the events, probabilities, and regions in your Venn diagram to avoid confusion.
• Check Your Work: Always double-check your calculations and make sure your answers make sense within the context of the problem.

Conclusion

Venn diagrams are an invaluable tool for understanding and solving probability problems. By visualizing the relationships between events, you can more easily calculate probabilities, especially when dealing with overlapping or conditional events. By mastering the techniques outlined in this article and practicing regularly, you'll be well-equipped to tackle a wide range of probability challenges. Remember to always read the problem carefully, draw a clear Venn diagram, fill it in systematically, and check your answers. With practice and attention to detail, you can unlock the power of Venn diagrams and become a probability pro.