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zimovet [89]
3 years ago
5

A certain disease occurs most frequently among older women. Of all age groups, women in their 60s have the highest rate of breas

t 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?
Mathematics
2 answers:
yuradex [85]3 years ago
7 0

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>A_k<em> given that another event, say X has already occurred is given by: </em>

         P(A_{k}|X)=\frac{P(X|A_{k})P(A_{k})}{\sum\limits^{k}_{i=1}{P(X|A_{i})P(A_{i})}}

Similarly,     P(BC/P) = \frac{P(BC) \times P(P/BC)}{P(BC) \times P(P/BC) + P(BC') \times P(P/BC')}

                 = \frac{0.0312 \times 0.81}{0.0312 \times 0.81 + 0.9688 \times 0.92}

                   = \frac{0.025272}{0.916568} = 0.0276

Hence, the required probability is 0.0276.

Law Incorporation [45]3 years ago
4 0

Answer:

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 <em>A</em>ₙ given that another event, say <em>X</em> has already occurred is given by:

P(A_{n}|X)=\frac{P(X|A_{n})P(A_{n})}{\sum\limits^{n}_{i=1}{P(X|A_{i})P(A_{i})}}

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

Denote the events as follows:

<em>B</em> = a women in their 60s has breast cancer

+ = the mammograms detects the breast cancer

The information provided is:

P(B) = 0.0312\\P(+|B)=0.81\\P(+|B^{c})=0.92

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

P(B|+)=\frac{P(+|B)P(B)}{P(+|B)P(B)+P(+|B^{c})P(B^{c})}

            =\frac{(0.81\times 0.0312)}{(0.81\times 0.0312)+(0.92\times (1-0.0312)}\\

            =\frac{0.025272}{0.025272+0.891296}

            =0.02757\\\approx0.0276

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

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