To find the probability that an individual has cancer given that the test result is negative, we need to use Bayes' theorem.

Let's define the terms in the problem with appropriate probabilities:

1. Sensitivity (True Positive Rate): The probability that the test is positive given that the individual has cancer, $P(\text{Positive} | \text{Cancer}) = 93.9\% = 0.939$.

2. Specificity (True Negative Rate): The probability that the test is negative given that the individual does not have cancer, $P(\text{Negative} | \text{No Cancer}) = 90.6\% = 0.906$.

3. Prevalence: The probability that an individual in this population has cancer, $P(\text{Cancer}) = \frac{1}{24} \approx 0.0417$.

We want to calculate the probability that an individual has cancer given that they tested negative, $P(\text{Cancer} | \text{Negative})$, which represents a false negative scenario.

To find this probability, we apply Bayes' theorem as follows:

$P(\text{Cancer} | \text{Negative}) = \frac{P(\text{Negative} | \text{Cancer}) \cdot P(\text{Cancer})}{P(\text{Negative})}$

Step 1: Calculate $P(\text{Negative} | \text{Cancer})$

Since sensitivity represents $P(\text{Positive} | \text{Cancer})$, the probability that a person with cancer tests negative is: $P(\text{Negative} | \text{Cancer}) = 1 - P(\text{Positive} | \text{Cancer}) = 1 - 0.939 = 0.061$

Step 2: Calculate $P(\text{Negative})$

The total probability of a negative result, $P(\text{Negative})$, is found by considering both cases: those with cancer and those without cancer. This is given by:

$P(\text{Negative}) = P(\text{Negative} | \text{Cancer}) \cdot P(\text{Cancer}) + P(\text{Negative} | \text{No Cancer}) \cdot P(\text{No Cancer})$

where:

• $P(\text{Negative} | \text{No Cancer}) = 0.906$,
• $P(\text{No Cancer}) = 1 - P(\text{Cancer}) = 1 - 0.0417 = 0.9583$.

Substituting the values, we get:

$P(\text{Negative}) = (0.061 \cdot 0.0417) + (0.906 \cdot 0.9583)$

Calculating each term:

• $0.061 \cdot 0.0417 = 0.0025457$,
• $0.906 \cdot 0.9583 = 0.8680098$.

Thus,

$P(\text{Negative}) = 0.0025457 + 0.8680098 = 0.8705555$

Step 3: Calculate $P(\text{Cancer} | \text{Negative})$

Now we substitute back into Bayes' theorem:

$P(\text{Cancer} | \text{Negative}) = \frac{P(\text{Negative} | \text{Cancer}) \cdot P(\text{Cancer})}{P(\text{Negative})} = \frac{0.061 \cdot 0.0417}{0.8705555}$

Calculating the numerator:

• $0.061 \cdot 0.0417 = 0.0025457$.

Then,

$P(\text{Cancer} | \text{Negative}) = \frac{0.0025457}{0.8705555} \approx 0.0029$

Final Answer

The probability that an individual has cancer given that they tested negative is approximately 0.29%.

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Would you like more detailed steps or clarification on any part of this solution?

Here are five related questions for further exploration:

1. What is the probability that an individual does not have cancer given a positive test result?
2. How would the probability change if the test's sensitivity were higher?
3. What is the probability of a false positive result with this test?
4. How would the prevalence of cancer affect the false negative rate?
5. How could one calculate the overall accuracy of this test?

Tip: Sensitivity and specificity can be remembered as measuring a test's "true positive" and "true negative" strengths, respectively.