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Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the probability P(X ≥ 14) given that n = 16 and p = 0.9. Round your answer to four decimal places.

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Final answer:

To find P(X ≥ 14) for a binomial distribution where n = 16 and p = 0.9, we sum the probabilities of getting 14, 15, and 16 successes, using either the binomial formula or a statistical calculator, and round to four decimal places.

Step-by-step explanation:

The subject of this question is finding the probability of a given number of successes in a binomial distribution, where X is the binomial random variable. In this case, we are asked to find the probability P(X ≥ 14) when n = 16 and the probability of success p = 0.9. To calculate this probability, one could either use the binomial probability formula to sum the probabilities of getting 14, 15, and 16 successes or use a calculator with binomial distribution functions. However, since this calculation can be labor-intensive, another common approach is to use the normal approximation for the binomial distribution, which is reasonable when np > 5 and nq > 5. Here, q = 1 - p.

For our specific problem, np = 16 x 0.9 = 14.4 and nq = 16 x (1 - 0.9) = 1.6. Since nq is not greater than 5, the approximation is not recommended. Therefore, we should calculate the exact probability using the binomial formula or a calculator. Assuming use of technology for calculation:

​P(X ≥ 14) = P(X = 14) + P(X = 15) + P(X = 16),
​and rounding our result to four decimal places as instructed.

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