To determine the probability that a patient with a negative mammogram actually has cancer, we use the concept of conditional probability. Specifically, we calculate $P(\text{Cancer} | \text{Negative Mammogram})$.

Let’s break it down systematically:

Step 1: Known Data (Assumptions)

• Prevalence of cancer ($P(\text{Cancer})$): Let’s call this $p_1$.
• False negative rate ($P(\text{Negative Mammogram} | \text{Cancer})$): Let’s call this $p_2$.
• Probability of no cancer ($P(\text{No Cancer})$): This is $1 - P(\text{Cancer})$.
• True negative rate ($P(\text{Negative Mammogram} | \text{No Cancer})$): Let’s call this $p_3$.

Step 2: Using Bayes' Theorem

We need to compute: $P(\text{Cancer} | \text{Negative Mammogram}) = \frac{P(\text{Negative Mammogram} | \text{Cancer}) \cdot P(\text{Cancer})}{P(\text{Negative Mammogram})}$

Step 3: Total Probability of a Negative Mammogram

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

Step 4: Plug Values into the Formula

You need to provide specific values for $P(\text{Cancer})$, $P(\text{Negative Mammogram} | \text{Cancer})$, and $P(\text{Negative Mammogram} | \text{No Cancer})$. Without these, I cannot compute the exact number. Could you provide these probabilities?