A survey found that $64 \%$ of adult Americans operates the flusher of toilets in public restrooms with their foot. Suppose a random sample of $n=500$ adult Americans is obtained and the number who flush public toilets with their foot is recorded. Find the shape of the binomial probability distribution. Since $n p(1-p)=$ the shape is Find the mean of the binomial probability distribution. adults who flush with their foot. Find the standard deviation of the binomial probability distribution. Round to three decimals adults who flush with their foot.

Step 1 ：The binomial distribution is defined by two parameters: the number of trials (n) and the probability of success in a single trial (p). In this case, the number of trials is the number of adult Americans surveyed (n=500) and the probability of success is the percentage of adult Americans who operate the flusher of toilets in public restrooms with their foot (p=0.64).

Step 2 ：The shape of a binomial distribution is determined by the values of n and p. If n is large and p is not too close to 0 or 1, the distribution will be approximately normal.

Step 3 ：The mean of a binomial distribution is given by the formula \(\mu = np\). Substituting the given values, we get \(\mu = 500 * 0.64 = 320\).

Step 4 ：The standard deviation of a binomial distribution is given by the formula \(\sigma = \sqrt{np(1-p)}\). Substituting the given values, we get \(\sigma = \sqrt{500 * 0.64 * (1 - 0.64)} = 10.733\), rounded to three decimal places.

Step 5 ：Final Answer: The shape of the binomial probability distribution is approximately normal because n is large and p is not too close to 0 or 1. The mean of the binomial probability distribution is \(\boxed{320}\) adults who flush with their foot. The standard deviation of the binomial probability distribution is \(\boxed{10.733}\) adults who flush with their foot.