Suppose that after reflecting on how difficult it is for people to interpret test results using probabilities and Bayes' Theorem, you try to improve people's reasoning by giving them the following information:

10 out of every 1,000 women at age 40 who participate in routine screening have breast cancer.

A of every 10 women with breast cancer will get a positive mammography. 95 out of every 990 women without breast cancer will also get a positive mammography. Here is a new representative sample of women at age 40 who got a positive mammography in routine screening. How many of these women do you expect to actually have breast cancer? blank out of blank

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.

Learn how to interpret medical test results using probabilities and Bayes' Theorem with a detailed example. This problem involves understanding how many women with a positive mammography actually have breast cancer, given specific probabilities and natural frequencies. Suitable for high school students and anyone interested in statistical reasoning.

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