An insurance company reported that $65 \%$ of all automobile damage claims were made by people under the age of 25 . If six automobile damage claims were selected at random, determine the probability that exactly two of them were made by someone under the age of 25.

The probability is $\square$
(Type an integer or decimal rounded to five decimal places as needed )

Step 1 ：This problem is a binomial probability problem. The binomial distribution model is appropriate here because we have a fixed number of trials (6 claims), each trial is independent (one claim does not affect another), there are only two outcomes (claim made by someone under 25 or not), and the probability of success (claim made by someone under 25) is the same for each trial (0.65).

Step 2 ：The formula for binomial probability is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\) where: \(P(X=k)\) is the probability of k successes in n trials, \(C(n, k)\) is the combination of n items taken k at a time, p is the probability of success, n is the number of trials, and k is the number of successes.

Step 3 ：In this case, n=6 (number of claims), k=2 (number of claims made by someone under 25), and p=0.65 (probability that a claim is made by someone under 25).

Step 4 ：Let's calculate the probability. The combination of 6 items taken 2 at a time is 15. The probability of 2 successes is \(0.65^2 = 0.4225\). The probability of 4 failures is \((1-0.65)^4 = 0.01500625\).

Step 5 ：The final probability is the product of these three values: \(15 * 0.4225 * 0.01500625 = 0.095102109375\).

Step 6 ：Final Answer: The probability that exactly two of them were made by someone under the age of 25 is approximately \(\boxed{0.09510}\).