4. Probability
Multiplication Rule: Independent Events
2:26 minutesProblem 3.2.14
Textbook Question
Classifying Events as Independent or Dependent In Exercises 9-14, determine whether the events are independent or dependent. Explain your reasoning.
14. A ball is selected from a bin of balls numbered from 1 through 52. It is replaced, and then a second numbered ball is selected from the bin.
Verified step by step guidance1
Understand the definition of independent and dependent events: Independent events are those where the outcome of one event does not affect the outcome of the other. Dependent events are those where the outcome of one event influences the outcome of the other.
Analyze the problem: A ball is selected from a bin of balls numbered 1 through 52, and then it is replaced before a second ball is selected. The replacement ensures that the total number of balls remains the same for both selections.
Consider the impact of replacement: Since the first ball is replaced, the probability of selecting any specific ball during the second draw remains the same as it was during the first draw. This indicates that the outcome of the first draw does not affect the second draw.
Conclude the relationship between the events: Because the replacement ensures that the conditions for the second draw are identical to the first, the two events are independent.
Explain the reasoning: The independence arises because the replacement resets the conditions, making the probability of selecting any ball during the second draw unaffected by the outcome of the first draw.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Independent Events
Independent events are those where the outcome of one event does not affect the outcome of another. In probability, two events A and B are independent if the probability of both occurring is the product of their individual probabilities, expressed as P(A and B) = P(A) * P(B). This concept is crucial for determining the relationship between events in probability scenarios.
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Dependent Events
Dependent events are those where the outcome of one event influences the outcome of another. For example, if the first event affects the sample space for the second event, they are considered dependent. The probability of both events occurring is calculated differently, as P(A and B) = P(A) * P(B|A), where P(B|A) is the probability of B given that A has occurred.
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Replacement in Probability
Replacement in probability refers to the practice of returning an item to the sample space after it has been selected. This action ensures that the probabilities remain constant for each selection. In the context of the given question, since the first ball is replaced before selecting the second, the events are independent, as the outcome of the first selection does not affect the second.
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