Define Independent And Mutually Exclusive
Define Independent And Mutually Ex
Please Complete Attachment Part 1 - Define independent and mutually exclusive events. Can two events be mutually exclusive and independent simultaneously? Support your answer with an example Part 2 - A bag contains 25 balls numbered 1 through 25. Suppose an odd number is considered a 'success'. Two balls are drawn from the bag with replacement. Find the probability of getting two successes, exactly one success, at least one success, no success Part 3 - Using the attached data set for our company, what is the probability of selecting each rank (for example, supervisor).
Paper For Above instruction
Introduction
Probability theory forms the foundation of decision-making processes under uncertainty, providing tools to evaluate the likelihood of various outcomes. Central to this theory are the concepts of independent events and mutually exclusive events, which describe different types of relationships between events and influence how probabilities are calculated and interpreted. Additionally, practical applications of probability, such as analyzing probabilities within real-world scenarios like drawing balls from a bag or selecting individuals with certain characteristics, further illustrate these concepts' relevance. This paper aims to define and distinguish between independent and mutually exclusive events, explore the possibility of their simultaneous occurrence, and apply probability calculations to specific scenarios involving a bag of balls and a company's dataset.
Part 1: Definitions and Relationship Between Independent and Mutually Exclusive Events
Mutually exclusive events, also known as disjoint events, are events that cannot occur simultaneously. Formally, two events A and B are mutually exclusive if their intersection is empty, which means P(A ∩ B) = 0. For example, when flipping a coin, the events "landing on heads" and "landing on tails" are mutually exclusive because the coin cannot land on both sides at once.
In contrast, independent events are events where the occurrence of one does not influence the probability of the other. Mathematically, events A and B are independent if P(A ∩ B) = P(A) × P(B). For example, if you roll a die and flip a coin, the outcome of the dice roll does not affect the coin flip, making these events independent.
A crucial relationship to note is that mutually exclusive events cannot be independent unless one of the events has a probability of zero. This is because, for mutually exclusive events A and B (with P(A ∩ B) = 0), if both have positive probabilities, then P(A) × P(B) > 0, which contradicts the fact that P(A ∩ B) = 0. Therefore, if two events are mutually exclusive and both have positive probabilities, they cannot be independent. Conversely, if two events are independent, they are not mutually exclusive unless one of the events is impossible (probability zero).
Can two events be mutually exclusive and independent simultaneously?
Yes, but only if at least one of the events has a probability of zero. For example, consider event A: rolling a 7 on a six-sided die (which is impossible, so P(A) = 0). Let event B be any event with P(B) > 0. Since P(A) = 0, P(A ∩ B) = 0, satisfying the independence condition P(A ∩ B) = P(A) × P(B) = 0. But realistically, mutually exclusive and independent events with positive probability for both are impossible because their intersection cannot be zero and equal to the product of their probabilities unless one has zero probability.
Part 2: Probability Calculations for Drawing Balls with Replacement
In this scenario, a bag contains 25 balls numbered 1 through 25, with odd numbers considered "successes." When two balls are drawn with replacement, the population remains unchanged because the balls are replaced after each draw.
Step 1: Probability of success (drawing an odd number)
There are 13 odd numbers between 1 and 25 (since 1, 3, 5, ..., 25).
Thus, P(success) = 13/25.
Step 2: Probability calculations for various outcomes
- Two successes: Both draws result in odd numbers.
Since draws are with replacement and the events are independent,
P(both successes) = P(success) × P(success) = (13/25) × (13/25) = 169/625 ≈ 0.2704.
- Exactly one success: One draw results in success, and the other does not.
There are two sequences: success then failure, or failure then success.
P(exactly one success) = 2 × P(success) × P(failure) = 2 × (13/25) × (12/25) = 2 × 156/625 = 312/625 ≈ 0.4992.
- At least one success: One or two successes.
P(at least one success) = 1 – P(no successes).
P(no successes) = P(failure) × P(failure) = (12/25) × (12/25) = 144/625 ≈ 0.2304.
Thus, P(at least one success) = 1 – 144/625 = 481/625 ≈ 0.7696.
- No success: Both draws result in failure (even numbers).
P(no success) = 144/625 ≈ 0.2304.
These calculations demonstrate how probability rules apply when dealing with repeated independent events with replacement, highlighting the importance of independence in such scenarios.
Part 3: Probabilities Based on Company Data
Using the provided dataset (assumed to include various employee ranks), the goal is to determine the probability of randomly selecting an individual with each rank—such as supervisor, manager, or worker.
Suppose the dataset contains 200 employees, with the following distribution:
- Supervisors: 40
- Managers: 30
- Employees: 130
The probability of selecting an individual of each rank is calculated by dividing the number of individuals in each category by the total number of employees.
- Probability of selecting a supervisor:
P(supervisor) = 40/200 = 0.20.
- Probability of selecting a manager:
P(manager) = 30/200 = 0.15.
- Probability of selecting a general employee:
P(employee) = 130/200 = 0.65.
These probabilities inform workforce analysis and decision-making, providing insights into the distribution of roles within the company. Understanding such probabilities enables HR and management to assess staff composition and strategize accordingly.
Conclusion
The distinction between independent and mutually exclusive events is fundamental in probability theory. While mutually exclusive events cannot occur simultaneously unless one of them is impossible, independent events have outcomes that do not influence each other. Recognizing when these concepts apply is crucial in practical scenarios, such as drawing balls from a bag or sampling employees from a dataset. Proper application of probability calculations enables precise estimations of various outcomes, facilitating better decision-making in business and research contexts. This understanding enhances the interpretive power of analyses involving chance, underpinning effective strategies across numerous fields.
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