To solve the problem, let's use the given data to determine how many women with a positive mammography actually have breast cancer. Here's the detailed calculation:

1. Women with breast cancer:

   • 10 out of 1,000 women have breast cancer.
   • Out of these 10 women, $\frac{9}{10}$ (90%) will get a positive mammography.
   • So, the number of women with breast cancer who get a positive mammography is $10 \times \frac{9}{10} = 9$.

2. Women without breast cancer:

   • 990 out of 1,000 women do not have breast cancer.
   • Out of these 990 women, $\frac{95}{990}$ (approximately 9.6%) will get a positive mammography.
   • So, the number of women without breast cancer who get a positive mammography is $990 \times \frac{95}{990} = 95$.

3. Total women with a positive mammography:

   • Women with breast cancer who test positive: 9
   • Women without breast cancer who test positive: 95
   • Total number of women who get a positive mammography is $9 + 95 = 104$.

4. Number of women with breast cancer among those with a positive mammography:

   • Out of the 104 women who get a positive mammography, 9 actually have breast cancer.

Thus, out of 104 women who got a positive mammography, 9 are expected to have breast cancer.

So, the answer is: 9 out of 104

Improving People's Reasoning:

Why this format might improve reasoning: Presenting information in natural frequencies (e.g., "10 out of 1,000 women") helps people understand probabilities more intuitively, as it aligns with everyday experiences and simplifies the mental calculations involved.

Limitation and Improvement: A limitation is that it still requires some mathematical manipulation to derive the final answer. This format could be improved by directly presenting the key probabilities or outcomes (e.g., "Among 104 women with a positive mammography, 9 will have breast cancer"), making it even easier for people to grasp the essential information without further calculation.

Do you want details or have any questions?

Here are some related questions you might ask:

1. Can you explain Bayes' Theorem in more detail?
2. How do you calculate probabilities from natural frequencies?
3. What are common misconceptions about interpreting medical test results?
4. How can Bayesian reasoning be applied in other contexts?
5. What are some other methods to improve statistical reasoning?

Tip: When interpreting statistical data, always try to break it down into simple, relatable terms to make the information more accessible and understandable.