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A binomial experiment with probability of success p= 0.68 and n= 9 trials is conducted. What is the probability that the experiment results in more than 6 successes? Do not round your intermediate computations, and round your answer to three decimal places. (If necessary, consult a list of formulas.) х 5 ?
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Step 1:: First, we need to determine the probability of success in a single trial, which is given as p = 0.68.
where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success in a single trial, and $\binom{n}{k}$ is the binomial coefficient.
Since this is a binomial experiment, the probability of success for any number of trials is given by the binomial probability formula:
Step 2:: To find the probability of more than 6 successes, we need to calculate the probabilities of exactly 6, 7, 8, and 9 successes and then sum them up.
Step 3:: Calculate the probability of exactly 6 successes:
P(X=6) = \binom{9}{6} (0.68)^6 (1-0.68)^{9-6}
Step 4:: Calculate the probability of exactly 7 successes:
P(X=7) = \binom{9}{7} (0.68)^7 (1-0.68)^{9-7}
Step 5:: Calculate the probability of exactly 8 successes:
P(X=8) = \binom{9}{8} (0.68)^8 (1-0.68)^{9-8}
Step 6:: Calculate the probability of exactly 9 successes:
P(X=9) = \binom{9}{9} (0.68)^9 (1-0.68)^{9-9}
Step 7:: Sum up the probabilities calculated in steps 3, 4, 5, and 6:
P(X>6) = P(X=6) + P(X=7) + P(X=8) + P(X=9)
Final Answer
P(X>6) \approx 0.152
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