Calculating Probability Of Event C In Independent Events A B And C
In probability theory, understanding the relationships between events is crucial for solving various problems. One such relationship is statistical independence, where the occurrence of one event does not affect the probability of another. This article delves into a specific scenario involving three statistically independent events, A, B, and C, defined over the same sample space S. We are given the probability of the union of these events, P(A ∪ B ∪ C), as well as the individual probabilities of events A and B, P(A) and P(B). Our goal is to compute the probability of event C, P(C). This problem highlights the application of fundamental probability concepts and formulas, particularly the inclusion-exclusion principle and the definition of statistical independence. By carefully applying these principles, we can derive a solution for P(C) that satisfies the given conditions.
Problem Statement
Let A, B, and C be three statistically independent events defined over the same sample space S. Assume P(A ∪ B ∪ C) = 0.92, P(A) = 0.2, and P(B) = 0.4. Compute P(C). The answer should be a number rounded to five decimal places.
Solution
To solve this problem, we will use the principle of inclusion-exclusion and the properties of statistically independent events. The inclusion-exclusion principle for three events states that:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)
Since A, B, and C are statistically independent, we have:
- P(A ∩ B) = P(A) * P(B)
- P(A ∩ C) = P(A) * P(C)
- P(B ∩ C) = P(B) * P(C)
- P(A ∩ B ∩ C) = P(A) * P(B) * P(C)
Substituting these into the inclusion-exclusion formula, we get:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A)P(B) - P(A)P(C) - P(B)P(C) + P(A)P(B)P(C)
We are given P(A ∪ B ∪ C) = 0.92, P(A) = 0.2, and P(B) = 0.4. Plugging these values into the equation, we have:
- 92 = 0.2 + 0.4 + P(C) - (0.2)(0.4) - (0.2)P(C) - (0.4)P(C) + (0.2)(0.4)P(C)
Simplifying the equation:
- 92 = 0.6 + P(C) - 0.08 - 0.2P(C) - 0.4P(C) + 0.08P(C)
Combining like terms:
- 92 = 0.52 + 0.48P(C)
Subtracting 0.52 from both sides:
- 4 = 0.48P(C)
Dividing by 0.48:
P(C) = 0.4 / 0.48
P(C) = 0.833333...
Rounding to five decimal places, we get:
P(C) ≈ 0.83333
Therefore, the probability of event C is approximately 0.83333.
Detailed Explanation of the Solution Steps
To fully grasp the solution, let's break down each step in detail:
1. Understanding Statistical Independence
The concept of statistical independence is fundamental to this problem. Two events are considered statistically independent if the occurrence of one does not influence the probability of the other. Mathematically, this is expressed as P(A ∩ B) = P(A) * P(B). This property extends to multiple events, allowing us to calculate the probability of intersections by multiplying the individual probabilities.
2. Applying the Inclusion-Exclusion Principle
The inclusion-exclusion principle is a crucial tool for calculating the probability of the union of events. It accounts for the overlaps between events to avoid double-counting. For three events, the principle is expressed as:
P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)
This formula ensures that we accurately calculate the probability of at least one of the events occurring.
3. Combining Independence and Inclusion-Exclusion
The key to solving this problem lies in combining the principle of inclusion-exclusion with the properties of statistical independence. By substituting the expressions for the intersections of independent events into the inclusion-exclusion formula, we obtain an equation that relates P(A ∪ B ∪ C), P(A), P(B), and P(C). This equation allows us to solve for the unknown probability, P(C).
4. Substituting Given Values
We are given P(A ∪ B ∪ C) = 0.92, P(A) = 0.2, and P(B) = 0.4. Substituting these values into the equation derived in the previous step, we get:
- 92 = 0.2 + 0.4 + P(C) - (0.2)(0.4) - (0.2)P(C) - (0.4)P(C) + (0.2)(0.4)P(C)
This equation now contains only one unknown, P(C), making it solvable.
5. Simplifying the Equation
The next step involves simplifying the equation by combining like terms. This includes adding the constant terms and grouping the terms involving P(C). The simplification process leads to a more manageable equation that can be easily solved for P(C).
6. Solving for P(C)
After simplifying the equation, we isolate P(C) by performing algebraic manipulations. This involves subtracting the constant term from both sides and then dividing by the coefficient of P(C). The result is the value of P(C), which represents the probability of event C occurring.
7. Rounding to Five Decimal Places
The problem specifies that the answer should be rounded to five decimal places. This is a common practice in numerical calculations to provide a precise and consistent result. We round the calculated value of P(C) to five decimal places to obtain the final answer.
Alternative Approaches
While the above solution is the most straightforward approach, there are alternative ways to solve this problem. One such approach involves using the complement rule.
Using the Complement Rule
The complement rule states that P(A') = 1 - P(A), where A' is the complement of event A. We can use this rule to find the probability of the complement of the union of events, P((A ∪ B ∪ C)'). Then, we can relate this to the complements of the individual events.
P((A ∪ B ∪ C)') = P(A' ∩ B' ∩ C')
Since A, B, and C are independent, their complements are also independent. Therefore:
P(A' ∩ B' ∩ C') = P(A') * P(B') * P(C')
We know that:
- P(A') = 1 - P(A) = 1 - 0.2 = 0.8
- P(B') = 1 - P(B) = 1 - 0.4 = 0.6
- P(A ∪ B ∪ C) = 0.92, so P((A ∪ B ∪ C)') = 1 - 0.92 = 0.08
Substituting these values, we get:
- 08 = 0.8 * 0.6 * P(C')
- 08 = 0.48 * P(C')
P(C') = 0.08 / 0.48 = 1/6
Now, we can find P(C) using the complement rule:
P(C) = 1 - P(C') = 1 - 1/6 = 5/6
P(C) ≈ 0.83333
This alternative approach yields the same result as the original method, providing a validation of our solution.
Common Mistakes to Avoid
When solving probability problems, it's crucial to avoid common mistakes that can lead to incorrect answers. Some of these mistakes include:
- Incorrectly applying the inclusion-exclusion principle: Forgetting to include or exclude terms in the formula can lead to significant errors.
- Assuming independence when it's not given: Events are not always independent, and incorrectly assuming independence can lead to wrong calculations.
- Double-counting probabilities: When calculating the probability of the union of events, it's important to account for overlaps to avoid double-counting.
- Making arithmetic errors: Simple arithmetic mistakes can propagate through the solution and lead to an incorrect final answer.
- Not rounding to the specified decimal places: Failing to round the answer to the required number of decimal places can result in a technically incorrect answer.
By being mindful of these common mistakes, you can improve your accuracy in solving probability problems.
Conclusion
In this article, we successfully computed the probability of event C, P(C), given the probabilities of the union of three statistically independent events A, B, and C, as well as the individual probabilities of events A and B. We utilized the inclusion-exclusion principle and the properties of statistical independence to derive the solution. We also explored an alternative approach using the complement rule, which validated our result. Understanding these concepts and techniques is essential for solving a wide range of probability problems. By carefully applying the principles and avoiding common mistakes, you can confidently tackle complex probability scenarios.
The calculated probability of event C is approximately 0.83333, rounded to five decimal places. This result demonstrates the power of probability theory in analyzing and understanding the relationships between events.
This problem serves as a valuable exercise in applying fundamental probability concepts and reinforces the importance of statistical independence and the inclusion-exclusion principle in solving probability-related questions. The detailed explanation and alternative approaches provided in this article offer a comprehensive understanding of the solution process and can aid in tackling similar problems in the future. Remember to always carefully consider the given information, apply the appropriate formulas, and avoid common mistakes to arrive at accurate solutions in probability problems.
