In the course of the discussion they noted the following fac

In

the

course

of

the

discussion

they

noted

the

following

facts

Only about 1 in 200 women in their 40\'s have breast cancer 86 2/3 percent of women will test positive if they have breast cancer There is 95 percent chance of a negative test result if a women don\'t have breast cancer Not all women get mammograms: Because of economic and social background women with breast cancer are less likely to get tested than women who do not have breast cancer 30% of women in their 40\'s with breast cancer get tested (have a mammogram) 40% of women in their 40\'s who do not have breast cancer get tested (have a mammogram) Based on this information answer the following questions (If the answer cannot be determined form the information provided: respond NA and explain what would be needed): The Probability (That a woman in her 40\'s gets tested) The Probability (That a woman in her 40\'s tests positive for breast cancer) The Probability (That a woman in her 40\'s has breast cancer given that she has tested positive) The Probability (That a woman in her 40\'s has breast cancer and does not get tested)- The Probability (That a woman in her 40\'s has breast cancer given that does not get tested) = If all women were tested for breast cancer the Probability (That a woman in her 40\'s would test positive)- Last Week thirteen women in their 40\'s were tested for breast cancer at the UB clinic. Of the first thirteen, 10 ten did not have breast cancer and three did have breast cancer. If we selected two of these women at random: 1. 2. 3. 4. 5. 6. 7. 8. What is the probability that both women had breast cancer? 9. What is the probability that both did not have breast cancer? 10. What is the probability that exactly one of women has breast cancer?

Solution

1 ) Probability that a woman in 40\'s get tested = (1/200)*30% + (199/200)*40%

= 39.95%

2)

Probability that the woman gets tested positive :

= (13.33%)*(1/200)*(0.30) + (5%)*(199/200)*(0.40)

= 2.01 %

3) Probability that she has breast cancer given that the test was positive :

= (13.33%)*(1/200)*(0.30) / 2.01%

= 2 / 201

4) Probability of having breast cancer and not getting tested:

= 70%

(i.e. 1 - probability of having breast cancer and getting tested)

5) Probability of having breast cancer given that it isn\'t tested

= [ ( 0.7 * 1/200) ] / [ ( 0.7 * 1/200) + (0.6 * 199/200) ]

= 7 / 1201

6) If all women were tested, positive results for breast cancer would be seen in:

= ( 260/3 * 1/200) + (199/200 * 5)

= 5.4083%

7)

a) choosing 2 out of 13, = ¹³C₂ ways = 78 ways

3 had breast cancer = ³C₂ ways = 3 ways

probability that both chosen had cancer = 3 /78 = 1 / 26

b)

choosing 2 out of 13, = ¹³C₂ ways = 78 ways

10 did not have breast cancer = ¹⁰C₂ ways = 45 ways

probability that both chosen did not have cancer = 45 /78 = 15 / 26

c)

choosing 2 out of 13, = ¹³C₂ ways = 78 ways

3 had breast cancer = ³C₁ ways = 3 ways

10 did not have breast cancer = ¹⁰C₁ ways = 10 ways

probability that exactly one had breast cancer =  ³C₁ *  ¹⁰C₁ /  ¹³C₂

= 30 / 78

= 5 / 13

Hope this helps.