The faculty of a community college were surveyed to determine the brand of smartphone they owned. The results are shown. If one of these individuals is selected at random, find the probability that they own a Brand B smartphone, given they are a woman. \begin{tabular}{|l|ccc|c|} \hline Faculty & \begin{tabular}{c} Brand \\ A \end{tabular} & \begin{tabular}{c} Brand \\ B \end{tabular} & \begin{tabular}{c} Brand \\ C \end{tabular} & Total \\ \hline Men & 35 & 25 & 50 & 110 \\ Women & 35 & 15 & 15 & 65 \\ \hline Total & 70 & 40 & 65 & 175 \\ \hline \end{tabular} $\mathrm{P}($ Brand $\mathrm{B} \mid$ woman $)=\square$ (Round to four decimal places as needed.)

Step 1 ：The problem is asking for the conditional probability of a woman owning a Brand B smartphone. The formula for conditional probability is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\). In this case, A is the event of owning a Brand B smartphone and B is the event of being a woman.

Step 2 ：\(P(A \cap B)\) is the probability of both A and B occurring, which is the number of women who own a Brand B smartphone divided by the total number of people. From the table, we can see that the number of women who own a Brand B smartphone is 15.

Step 3 ：\(P(B)\) is the probability of being a woman, which is the number of women divided by the total number of people. From the table, we can see that the total number of women is 65.

Step 4 ：Substitute these values into the formula, we get \(P(A|B) = \frac{15}{65} = 0.23076923076923078\).

Step 5 ：Rounding to four decimal places, the probability of a woman owning a Brand B smartphone is approximately 0.2308.

Step 6 ：Final Answer: \(\boxed{0.2308}\)