Consider the following binomial experiment:
According to a survey, $64 \%$ of adult Americans operate the flusher of toilets in public restrooms with their foot. Suppose a random sample of $n=20$ adult Americans is obtained and the number who flush public toilets with their foot is recorded.
What is the probability that exactly 13 flush public toilets with their foot?
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What is the probability that at least 15 flush public toilets with their foot?
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Step 1 ：We are given a binomial distribution problem where the number of trials is 20 (n=20), the probability of success is 0.64 (p=0.64), and we want to find the probability of exactly 13 successes (k=13).

Step 2 ：The formula for the binomial distribution is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\), where C(n, k) is the number of combinations of n items taken k at a time.

Step 3 ：First, we calculate the number of combinations of 20 items taken 13 at a time, which is 77520.

Step 4 ：Next, we calculate \(p^k\), which is \(0.64^{13}\) and equals approximately 0.003022314549036574.

Step 5 ：Then, we calculate \((1-p)^(n-k)\), which is \((1-0.64)^{20-13}\) and equals approximately 0.0007836416409599998.

Step 6 ：Finally, we substitute these values into the binomial distribution formula to get the probability: \(P(X=13) = 77520 * 0.003022314549036574 * 0.0007836416409599998\), which equals approximately 0.18359926201523752.

Step 7 ：Rounding to three decimal places, the probability that exactly 13 out of 20 people flush public toilets with their foot is approximately \(\boxed{0.184}\).