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 \(\boxed{0.184}\).