4. Probability
Addition Rule
1:43 minutesProblem 3.3.1
Textbook Question
1. When two events are mutually exclusive, why is P(A and B) = 0?
Verified step by step guidance1
Understand the concept of mutually exclusive events: Two events are mutually exclusive if they cannot occur at the same time. For example, flipping a coin and getting both heads and tails simultaneously is impossible, making these events mutually exclusive.
Recall the definition of the intersection of events: The intersection of two events, denoted as P(A and B), represents the probability that both events A and B occur simultaneously.
Apply the property of mutually exclusive events: Since mutually exclusive events cannot happen at the same time, the intersection of these events is empty. Mathematically, this means P(A and B) = 0.
Visualize the concept using a Venn diagram: In a Venn diagram, mutually exclusive events do not overlap. The absence of overlap visually confirms that P(A and B) = 0.
Conclude with the reasoning: The probability of two mutually exclusive events occurring together is zero because their definitions inherently prevent simultaneous occurrence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mutually Exclusive Events
Mutually exclusive events are those that cannot occur simultaneously. If one event happens, the other cannot. For example, when flipping a coin, getting heads and tails at the same time is impossible. This concept is crucial for understanding why the probability of both events occurring together is zero.
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Probability of Events
Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. The probability of two events occurring together, denoted as P(A and B), is calculated based on their individual probabilities. For mutually exclusive events, since they cannot happen at the same time, the probability of both occurring is always zero.
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Addition Rule of Probability
The addition rule of probability states that for any two events A and B, the probability of either A or B occurring is the sum of their individual probabilities, minus the probability of both occurring together. For mutually exclusive events, since P(A and B) = 0, the rule simplifies to P(A or B) = P(A) + P(B), reinforcing that they cannot happen at the same time.
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