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P(X ≤ 1). n = 5, p = 0.6

P(X ≤ 1). n = 5, p = 0.6

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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### Binomial Probability Calculation **Given:** - \(P(X \leq 1)\) - \(n = 5\) - \(p = 0.6\) This mathematical expression represents the cumulative probability in a binomial distribution. - **\(P(X \leq 1)\)**: This indicates the probability that the random variable \(X\) is less than or equal to 1. - **\(n = 5\)**: This represents the number of trials or experiments. - **\(p = 0.6\)**: This denotes the probability of success in each trial. In the context of a binomial distribution, we are interested in finding the probability that \(X\), the number of successes in 5 trials, is at most 1 when the probability of success in each trial is 0.6. To calculate this cumulative probability, we typically use the binomial probability formula or statistical software/table: \[ P(X \leq 1) = \sum_{k=0}^{1} {P(X = k)} \] where: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Here, \(\binom{n}{k}\) is the binomial coefficient. The probability values can be summed up for \(k = 0\) and \(k = 1\): \[ P(X \leq 1) = P(X = 0) + P(X = 1) \] By calculating each individual term, one can find the cumulative probability for \(X \leq 1\).
Transcribed Image Text:### Binomial Probability Calculation **Given:** - \(P(X \leq 1)\) - \(n = 5\) - \(p = 0.6\) This mathematical expression represents the cumulative probability in a binomial distribution. - **\(P(X \leq 1)\)**: This indicates the probability that the random variable \(X\) is less than or equal to 1. - **\(n = 5\)**: This represents the number of trials or experiments. - **\(p = 0.6\)**: This denotes the probability of success in each trial. In the context of a binomial distribution, we are interested in finding the probability that \(X\), the number of successes in 5 trials, is at most 1 when the probability of success in each trial is 0.6. To calculate this cumulative probability, we typically use the binomial probability formula or statistical software/table: \[ P(X \leq 1) = \sum_{k=0}^{1} {P(X = k)} \] where: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Here, \(\binom{n}{k}\) is the binomial coefficient. The probability values can be summed up for \(k = 0\) and \(k = 1\): \[ P(X \leq 1) = P(X = 0) + P(X = 1) \] By calculating each individual term, one can find the cumulative probability for \(X \leq 1\).
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