Step 1 :We are given that the number of trials \(n = 10\), the probability of success \(p = 0.65\), and the number of successes \(x = 6\).
Step 2 :We use the binomial probability formula to calculate the probability of getting exactly 6 successes in 10 trials. The formula is \(P(x) = C(n, x) \cdot p^x \cdot (1-p)^{n-x}\), where \(C(n, x)\) is the number of combinations of \(n\) items taken \(x\) at a time.
Step 3 :We calculate the number of combinations \(C(n, x) = C(10, 6) = 210\).
Step 4 :We calculate \(p^x = 0.65^6 = 0.07541889062500001\).
Step 5 :We calculate \((1-p)^{n-x} = (1-0.65)^{10-6} = 0.015006249999999997\).
Step 6 :We substitute these values into the binomial probability formula to get the probability \(P(6) = 210 \cdot 0.07541889062500001 \cdot 0.015006249999999997 = 0.2376684927626953\).
Step 7 :We round the final answer to four decimal places to get \(\boxed{0.2377}\).