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What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1?
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What is the expectation of the number of failures preceding the first ...
Introduction
In this scenario, we are analyzing the number of failures before the first success in a series of independent Bernoulli trials, where each trial has a success probability of e^(-1).
Understanding the Problem
- Each trial has two outcomes: success (with probability p = e^(-1)) and failure (with probability q = 1 - p = 1 - e^(-1)).
- We are interested in finding the expected number of failures before the first success.
Geometric Distribution
- The number of trials until the first success follows a geometric distribution.
- The expected number of failures (X) preceding the first success can be calculated using the formula: E[X] = q/p.
Calculating the Expectation
- We know:
- p = e^(-1)
- q = 1 - e^(-1)
- Substituting the values into the expectation formula:
E[X] = (1 - e^(-1)) / e^(-1)
- Simplifying this gives:
E[X] = (1 - e^(-1)) * (e) = e - 1.
Conclusion
- Therefore, the expected number of failures preceding the first success when the success probability is e^(-1) is e - 1.
- This result illustrates the relationship between the probabilities of success and failure in a geometric distribution, highlighting how the expectation increases as the probability of success decreases.
In this scenario, we are analyzing the number of failures before the first success in a series of independent Bernoulli trials, where each trial has a success probability of e^(-1).
Understanding the Problem
- Each trial has two outcomes: success (with probability p = e^(-1)) and failure (with probability q = 1 - p = 1 - e^(-1)).
- We are interested in finding the expected number of failures before the first success.
Geometric Distribution
- The number of trials until the first success follows a geometric distribution.
- The expected number of failures (X) preceding the first success can be calculated using the formula: E[X] = q/p.
Calculating the Expectation
- We know:
- p = e^(-1)
- q = 1 - e^(-1)
- Substituting the values into the expectation formula:
E[X] = (1 - e^(-1)) / e^(-1)
- Simplifying this gives:
E[X] = (1 - e^(-1)) * (e) = e - 1.
Conclusion
- Therefore, the expected number of failures preceding the first success when the success probability is e^(-1) is e - 1.
- This result illustrates the relationship between the probabilities of success and failure in a geometric distribution, highlighting how the expectation increases as the probability of success decreases.
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What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1?
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What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1?.
What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What is the expectation of the number of failures preceding the first success in an infinite series of independent trials with constant probability e^(-1) of success in each trial? How is the answer e-1?.
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