Problem

Suppose that X is a random variable with the binomial distribution with n=18 and p=0.8453. Calculate P(X=14). If rounding is necessary, round as indicated on the formula sheet.

Answer

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Answer

Final Answer: The probability P(X=14) is approximately 0.1667.

Steps

Step 1 :We are given that X is a random variable with the binomial distribution with n=18 and p=0.8453. We are asked to calculate P(X=14).

Step 2 :The binomial distribution is given by the formula: P(X=k)=(nk)pk(1p)nk, where n is the number of trials, k is the number of successes, p is the probability of success on each trial, and (nk) is the binomial coefficient, which can be calculated as n!k!(nk)!.

Step 3 :Substitute n=18, k=14, and p=0.8453 into the formula to calculate P(X=14).

Step 4 :Calculate the binomial coefficient (nk): (1814)=3060.0.

Step 5 :Substitute the binomial coefficient and the given values into the formula: P(X=14)=3060.00.845314(10.8453)1814.

Step 6 :Solve the equation to get the probability: P(X=14)=0.16666133140213757.

Step 7 :Round the result to four decimal places: P(X=14)0.1667.

Step 8 :Final Answer: The probability P(X=14) is approximately 0.1667.

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