Find The Probability Of Obtaining At Least One 1 When A Single Fair Die Is Rolled Three Times.Make An Argument Using A Venn Diagram To Justify The General Addition Rule: ${ P(E \cup F) = P(E) + P(F) - P(E \cap F) }$

Introduction

When a single fair die is rolled three times, we are interested in finding the probability of obtaining at least one "1". This problem can be approached using the General Addition Rule, which states that the probability of the union of two events E and F is given by:

${ P(E \cup F) = P(E) + P(F) - P(E \cap F) }$

In this article, we will use a Venn diagram to justify the General Addition Rule and find the probability of obtaining at least one "1" when a single fair die is rolled three times.

The General Addition Rule

The General Addition Rule is a fundamental concept in probability theory that allows us to find the probability of the union of two events. The rule states that the probability of the union of two events E and F is given by:

${ P(E \cup F) = P(E) + P(F) - P(E \cap F) }$

This rule can be visualized using a Venn diagram, which is a diagram that shows the relationships between sets.

Venn Diagram

A Venn diagram is a diagram that shows the relationships between sets. In the context of the General Addition Rule, a Venn diagram can be used to visualize the union of two events E and F.

+---------------+
|               |
|  E            |
|               |
+---------------+
       |
       |
       v
+---------------+
|               |
|  F            |
|               |
+---------------+
       |
       |
       v
+---------------+
|               |
|  E ∩ F       |
|               |
+---------------+

In this Venn diagram, the region labeled E represents the event E, the region labeled F represents the event F, and the region labeled E ∩ F represents the intersection of the two events.

Justifying the General Addition Rule

Using the Venn diagram, we can justify the General Addition Rule as follows:

• The probability of the union of the two events E and F is equal to the sum of the probabilities of the two events minus the probability of their intersection.
• The probability of the intersection of the two events E and F is equal to the area of the region labeled E ∩ F.
• The probability of the union of the two events E and F is equal to the sum of the areas of the regions labeled E and F minus the area of the region labeled E ∩ F.

Mathematically, this can be represented as:

${ P(E \cup F) = P(E) + P(F) - P(E \cap F) }$

Finding the Probability of Obtaining at Least One "1"

Now that we have justified the General Addition Rule using a Venn diagram, we can use it to find the probability of obtaining at least one "1" when a single fair die is rolled three times.

Let E be the event that the first roll is a "1", F be the event that the second roll is a "1", and G be the event that the third roll is a "1".

We are interested in finding the probability of the union of the three events E, F, and G, which is given by:

${ P(E \cup F \cup G) = P(E) + P(F) + P(G) - P(E \cap F) - P(F \cap G) - P(G \cap E) + P(E \cap F \cap G) }$

Since the die is fair, the probability of obtaining a "1" on any roll is 1/6.

Therefore, the probability of obtaining at least one "1" when a single fair die is rolled three times is:

${ P(E \cup F \cup G) = 3 \times \frac{1}{6} - 2 \times \frac{1}{6} \times \frac{1}{6} + \frac{1}{6} \times \frac{1}{6} \times \frac{1}{6} }$

Simplifying the expression, we get:

${ P(E \cup F \cup G) = \frac{1}{2} - \frac{1}{36} + \frac{1}{216} }$

${ P(E \cup F \cup G) = \frac{108 - 1 + 1}{216} }$

${ P(E \cup F \cup G) = \frac{108}{216} }$

${ P(E \cup F \cup G) = \frac{1}{2} }$

Therefore, the probability of obtaining at least one "1" when a single fair die is rolled three times is 1/2 or 50%.

Conclusion

In this article, we used a Venn diagram to justify the General Addition Rule and find the probability of obtaining at least one "1" when a single fair die is rolled three times. We showed that the probability of obtaining at least one "1" is 1/2 or 50%. This result can be used to make informed decisions in a variety of situations, such as in games of chance or in statistical analysis.

References

• Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
• Ross, S. M. (2014). A First Course in Probability. Pearson Education.

Further Reading

• Probability Theory: A comprehensive introduction to probability theory, including the General Addition Rule.
• Venn Diagrams: A tutorial on how to use Venn diagrams to visualize the relationships between sets.
• Probability and Statistics: A collection of resources on probability and statistics, including tutorials, examples, and exercises.
  

Q: What is the General Addition Rule?

A: The General Addition Rule is a fundamental concept in probability theory that allows us to find the probability of the union of two events. The rule states that the probability of the union of two events E and F is given by:

${ P(E \cup F) = P(E) + P(F) - P(E \cap F) }$

Q: How is the General Addition Rule used to find the probability of obtaining at least one "1" when a single fair die is rolled three times?

A: To find the probability of obtaining at least one "1" when a single fair die is rolled three times, we use the General Addition Rule to find the probability of the union of the three events E, F, and G, where E is the event that the first roll is a "1", F is the event that the second roll is a "1", and G is the event that the third roll is a "1".

Q: What is the probability of obtaining a "1" on any roll of a fair die?

A: The probability of obtaining a "1" on any roll of a fair die is 1/6.

Q: How do we calculate the probability of obtaining at least one "1" when a single fair die is rolled three times?

A: To calculate the probability of obtaining at least one "1" when a single fair die is rolled three times, we use the formula:

${ P(E \cup F \cup G) = P(E) + P(F) + P(G) - P(E \cap F) - P(F \cap G) - P(G \cap E) + P(E \cap F \cap G) }$

Q: What is the probability of obtaining at least one "1" when a single fair die is rolled three times?

A: The probability of obtaining at least one "1" when a single fair die is rolled three times is 1/2 or 50%.

Q: Can you explain the Venn diagram used to justify the General Addition Rule?

A: The Venn diagram used to justify the General Addition Rule is a diagram that shows the relationships between sets. In the context of the General Addition Rule, the Venn diagram can be used to visualize the union of two events E and F.

+---------------+
|               |
|  E            |
|               |
+---------------+
       |
       |
       v
+---------------+
|               |
|  F            |
|               |
+---------------+
       |
       |
       v
+---------------+
|               |
|  E ∩ F       |
|               |
+---------------+

Q: What are some real-world applications of the General Addition Rule?

A: The General Addition Rule has many real-world applications, including:

• Games of chance: The General Addition Rule can be used to calculate the probability of winning a game of chance, such as a lottery or a casino game.
• Statistical analysis: The General Addition Rule can be used to calculate the probability of a certain event occurring in a statistical analysis.
• Risk management: The General Addition Rule can be used to calculate the probability of a certain event occurring in a risk management analysis.

Q: What are some common mistakes to avoid when using the General Addition Rule?

A: Some common mistakes to avoid when using the General Addition Rule include:

• Not accounting for the intersection of events: The General Addition Rule requires that we account for the intersection of events, which can be a common mistake.
• Not using the correct formula: The General Addition Rule requires that we use the correct formula, which can be a common mistake.
• Not considering the probability of the union of events: The General Addition Rule requires that we consider the probability of the union of events, which can be a common mistake.

Q: How can I practice using the General Addition Rule?

A: You can practice using the General Addition Rule by working through examples and exercises, such as:

• Calculating the probability of obtaining at least one "1" when a single fair die is rolled three times: This is a classic example of using the General Addition Rule.
• Calculating the probability of winning a game of chance: This is a real-world application of the General Addition Rule.
• Calculating the probability of a certain event occurring in a statistical analysis: This is a real-world application of the General Addition Rule.

Q: What are some resources for learning more about the General Addition Rule?

A: Some resources for learning more about the General Addition Rule include:

• Probability Theory: A comprehensive introduction to probability theory, including the General Addition Rule.
• Venn Diagrams: A tutorial on how to use Venn diagrams to visualize the relationships between sets.
• Probability and Statistics: A collection of resources on probability and statistics, including tutorials, examples, and exercises.