A bag contains five red marbles and eleven white marbles. If a sample of five marbles contains at least one white marble, what is the probability that all the marbles in the sample are white?

The probability that all the marbles in a sample of five marbles are white given that at least one of the marbles is white is $\square$. (Round to four decimal places as needed.)

Step 1 ：The problem is asking for the conditional probability that all five marbles drawn are white given that at least one marble drawn is white. To solve this, we first need to calculate the total number of ways to draw five marbles from the bag, the number of ways to draw at least one white marble, and the number of ways to draw five white marbles. Then we can use these values to calculate the conditional probability.

Step 2 ：The total number of ways to draw five marbles from the bag is given by the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n\) is the total number of marbles and \(k\) is the number of marbles drawn. In this case, \(n = 5 + 11 = 16\) and \(k = 5\). So, the total number of ways to draw five marbles is 4368.

Step 3 ：The number of ways to draw at least one white marble is the total number of ways to draw five marbles minus the number of ways to draw five red marbles. The number of ways to draw five red marbles is given by the combination formula with \(n = 5\) and \(k = 5\). So, the number of ways to draw at least one white marble is \(4368 - 1 = 4367\).

Step 4 ：The number of ways to draw five white marbles is given by the combination formula with \(n = 11\) and \(k = 5\). So, the number of ways to draw five white marbles is 462.

Step 5 ：Finally, the conditional probability is given by the ratio of the number of ways to draw five white marbles to the number of ways to draw at least one white marble. So, the probability is \(\frac{462}{4367} = 0.10579345088161209\).

Step 6 ：Final Answer: The probability that all the marbles in a sample of five marbles are white given that at least one of the marbles is white is \(\boxed{0.1058}\). (Round to four decimal places as needed.)