To solve this, we need to use Bayes' theorem. First, let’s identify the key probabilities:

• $P(C)$: The probability of having cancer (prior probability).
• $P(\neg C)$: The probability of not having cancer.
• $P(\neg M \mid C)$: The probability of a negative mammogram given cancer.
• $P(\neg M \mid \neg C)$: The probability of a negative mammogram given no cancer.
• $P(\neg M)$: The overall probability of a negative mammogram.

We want to calculate:

$P(C \mid \neg M) = \frac{P(\neg M \mid C) P(C)}{P(\neg M)}$

Steps:

1. Find $P(\neg M)$:

Using the law of total probability: $P(\neg M) = P(\neg M \mid C)P(C) + P(\neg M \mid \neg C)P(\neg C)$

2. Substitute the values and compute $P(C \mid \neg M)$:

Provide the specific probabilities, and I will calculate the result. If you don’t have the probabilities, clarify the following so I can proceed:

• What is $P(C)$ (the prevalence of cancer)?
• What is $P(\neg M \mid C)$ (the sensitivity of the test)?
• What is $P(\neg M \mid \neg C)$ (the specificity of the test)?

Let me know, and I’ll finalize the calculation.