A certain disease occurs most frequently among older women. Of all age groups, women in their 60s have the highest rate of breast cancer. The National Cancer Institute (NCI) compiles U. S. epidemiology data for a number of different cancers. The NCI estimates that 3.12% of women in their 60s get breast cancer. Mammograms are X-ray images of the breast used to detect breast cancer. A mammogram can typically identify correctly 81% of cancer cases and 92% of cases without cancer. If a woman in her 60s gets a positive mammogram, what is the probability that she indeed has breast cancer?

The probability that a woman in her 60s has breast cancer given that she gets a positive mammogram is 0.0276.

Step-by-step explanation:

Let a set be events that have occurred be denoted as:

S = {A₁, A₂, A₃,..., Aₙ}

The Bayes' theorem states that the conditional probability of an event, say Aₙ given that another event, say X has already occurred is given by:

The disease Breast cancer is being studied among women of age 60s.

Denote the events as follows:

B = a women in their 60s has breast cancer

+ = the mammograms detects the breast cancer

The information provided is:

Compute the value of P (B|+) using the Bayes' theorem as follows:

Thus, the probability that a woman in her 60s has breast cancer given that she gets a positive mammogram is 0.0276.