Answer:
The probability that she indeed has breast cancer is 0.0276.
Step-by-step explanation:
We are given that a certain disease occurs most frequently among older women. Of all age groups, women in their 60's have the highest rate of breast cancer.
The NCI estimates that 3.12% of women in their 60's get breast cancer. A mammogram can typically identify correctly 81% of cancer cases and 92% of cases without cancer.
Let Probability that women in their 60's get breast cancer = P(BC) = 0.0312
Probability that women in their 60's does not get breast cancer = P(BC') = 1 - P(BC) = 1 - 0.0312 = 0.9688
Also, let P = event that mammograms correctly detect positive results for breast cancer
So, Probability that mammograms correctly detect positive results given that women actually has breast cancer = P(P/BC) = 0.81
Probability that mammograms detect correctly given that women actually does not has breast cancer = P(P/BC') = 0.92
Now, to find the probability that she indeed has breast cancer given the fact that woman in her 60's gets a positive mammogram, we will use Bayes' Theorem;
The Bayes' theorem is given by;
<em>The Bayes' theorem states that the conditional probability of an event, say </em><em> given that another event, say X has already occurred is given by:
</em>
Similarly, P(BC/P) =
=
= = 0.0276
Hence, the required probability is 0.0276.