Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, the trials are the selection of civilian employees, and a 'success' is defined as an employee having access to medical care benefits. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).
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Probability Calculation
To find the probability of a certain number of successes in a binomial distribution, we use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k). Here, 'n choose k' represents the number of ways to choose k successes from n trials, p is the probability of success, and (1-p) is the probability of failure. This formula allows us to calculate the likelihood of observing a specific number of employees with benefits.
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Cumulative Probability
Cumulative probability refers to the probability of obtaining a value less than or equal to a certain number in a distribution. In this case, to find the probability that at least six employees have access to medical care benefits, we can calculate the cumulative probabilities for six, seven, eight, and nine employees and sum these values. This approach is essential for determining probabilities for ranges of outcomes in binomial scenarios.
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