6. In a certain region of the country, it is known from past experience that the
probability of selecting an adult over 40 years of age with cancer is 0.05. If the
probability of a doctor correctly diagnosing a person with cancer as having the
disease is 0.78 and the probability of incorrectly diagnosing a person without
cancer as having the disease is 0.06, what is the probability that an adult over 40
years of age is diagnosed as having cancer?

To solve this problem, we need to calculate the probability that an adult over 40 years of age is diagnosed as having cancer, considering both the correct and incorrect diagnoses. This involves applying the law of total probability.

### Given Data:
- \( P(C) = 0.05 \): The probability that an adult over 40 years has cancer.
- \( P(D|C) = 0.78 \): The probability that a doctor correctly diagnoses a person with cancer.
- \( P(D|\neg C) = 0.06 \): The probability that a doctor incorrectly diagnoses a person without cancer as having the disease.
- \( P(\neg C) = 1 - P(C) = 0.95 \): The probability that an adult over 40 years does not have cancer.

### Probability of Diagnosis:
We want to find the probability that an adult over 40 years is diagnosed as having cancer, which is \( P(D) \). This can be found using the law of total probability:

\[
P(D) = P(D|C)P(C) + P(D|\neg C)P(\neg C)
\]

Substituting the given values:

\[
P(D) = (0.78 \times 0.05) + (0.06 \times 0.95)
\]

Now, let's calculate:

\[
P(D) = (0.78 \times 0.05) + (0.06 \times 0.95)
\]
\[
P(D) = 0.039 + 0.057 = 0.096
\]

### Final Answer:
The probability that an adult over 40 years of age is diagnosed as having cancer is **0.096** or **9.6%**.

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Would you like more details or have any questions?

Here are five related questions to expand on this topic:

1. What is the probability that an adult over 40 years of age does not have cancer but is diagnosed as having it?
2. How would the probability change if the prevalence of cancer in the region increased to 0.10?
3. What is the probability that an adult over 40 years is correctly diagnosed as not having cancer?
4. How would the probability of diagnosis change if the doctor's diagnostic accuracy improved?
5. How would you calculate the probability that an adult over 40 years of age who is diagnosed with cancer actually has the disease?

**Tip:** When dealing with conditional probabilities, always carefully define your events and their respective probabilities before applying any formulas.

Explore the probability of diagnosing cancer in adults over 40 in a specific region. Learn how to calculate the probability using conditional probability and the law of total probability. Includes detailed steps and calculations based on given data.

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