Suppose that $X$ is a random variable with the binomial distribution with $n=16$ and $p=0.6853$. Calculate $P(X=10)$. If rounding is necessary, round as indicated on the formula sheet.

Step 1 ：We are given that $X$ is a random variable with the binomial distribution with $n=16$ and $p=0.6853$. We are asked to calculate $P(X=10)$.

Step 2 ：The binomial distribution is given by the formula: \[P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\] where: $n$ is the number of trials, $k$ is the number of successful trials, $p$ is the probability of success on each trial, and $\binom{n}{k}$ is the binomial coefficient, which can be calculated as $\frac{n!}{k!(n-k)!}$.

Step 3 ：In this case, we have $n=16$, $k=10$, and $p=0.6853$. We can substitute these values into the formula to find $P(X=10)$.

Step 4 ：First, we calculate the binomial coefficient $\binom{n}{k}$, which is $\binom{16}{10} = 8008.0$.

Step 5 ：Next, we substitute $n=16$, $k=10$, $p=0.6853$, and $\binom{n}{k}=8008.0$ into the formula to get $P(X=10) = 8008.0 \cdot (0.6853)^{10} \cdot (1-0.6853)^{16-10} = 0.17770998905405114$.

Step 6 ：Final Answer: The probability $P(X=10)$ is approximately \(\boxed{0.1777}\).