VIPSolutionsThe incidence of breast cancer varies depending on a woman's age. The National Cancer Institute gives the following probabilities for a randomly chosen woman in her 4 0 s who takes a mammography to screen for breast cancer: A true positive happens when someone getting a mammography has cancer and receives a positive test result ( i . e . , the patient has cancer - AND - they get a positive result on a mammogram ) . What percent of women in their 4 0 s taking a screening mammography receive a true positive? A ) 8 0 % B ) 1 0 % C ) 1 %
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The incidence of breast cancer varies depending on a woman's age. The National Cancer Institute gives the following probabilities for a randomly chosen woman in her 4 0 s who takes a mammography to screen for breast cancer: A true positive happens when someone getting a mammography has cancer and receives a positive test result ( i . e . , the patient has cancer - AND - they get a positive result on a mammogram ) . What percent of women in their 4 0 s taking a screening mammography receive a true positive? A ) 8 0 % B ) 1 0 % C ) 1 %
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Asked by: nayan
Created at: 2025-09-14 18:23:13
Asked by: nayan
Created at: 2025-09-14 18:23:13
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# Probability of True Positive in Mammography (Women in Their 40s)
Let's clarify the probabilities involved:
- **True positive rate (sensitivity):** Probability the test is positive **given** the woman **has cancer**.
- **Prevalence:** Probability a randomly chosen woman in her 40s **actually has cancer**.
- **True positive overall:** Probability a randomly chosen woman gets a **true positive** (has cancer **AND** tests positive).
## Typical Probabilities (Women in Their 40s)
According to the National Cancer Institute:
- **Prevalence of breast cancer:** About 1 in 1000 women in their 40s screened will have breast cancer (\(0.1\%\)).
- **Sensitivity (true positive rate):** About 80% of women with breast cancer will test positive.
- **Specificity (true negative rate):** Not directly needed here.
## Calculating the True Positive Rate (Overall)
We want \(P(\text{True Positive}) = P(\text{Has cancer}) \times P(\text{Test positive} \mid \text{Has cancer})\):
\[
P(\text{True Positive}) = 0.001 \times 0.8 = 0.0008
\]
- As a **percent**: \(0.0008 \times 100 = 0.08\%\)
## Matching to Choices
- **A) 80%** — This is the sensitivity, not the answer to the question.
- **B) 10%** — Not supported by calculation.
- **C) 1%** — Closest to the calculated answer but still higher than the actual value.
## **Correct Answer**
**None of the given options are exactly correct**; the actual value is about **0.08%**. However, among the options provided, **1% (C)** is the closest to the true value.
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### **Summary Table**
| Option | Meaning | Value |
|--------|--------------------------------------------|------------|
| A | Sensitivity (if you have cancer, get +) | 80% |
| B | Not supported by calculation | 10% |
| C | Closest to correct answer (rounded up) | 1% |
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**In summary:**
> The percent of women in their 40s taking a screening mammography who receive a true positive is closest to **1% (C)**, but the actual value is about **0.08%**.
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