Ams 310 Survey Of Probability And Statistics Worksheet - Midterm Exam I - Stony Brook University Page 6
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11(4). (a). About 40 percent of women who take pregnancy test are actually carrying
a baby. The testing is 99 percent accurate. If a woman takes a test and
the result is positive (showing pregnancy), what is the probability that she is
actually carrying a baby?
(b). Among all women who take mammography test each year, only 0.1 percent
of them actually have breast cancer. Suppose the mammography is 99 per-
cent accurate and a woman is tested positive for breast cancer, what is the
probability she actually does not have cancer?
Solution:
(a). Let
be the event of non-pregnant woman taking a test and
be the event
¯ A
A
of pregnant woman taking a test, we have
and
. Let
P ( ¯ A) = 0.4
P (A) = 0.6
B
denote the event that the test is positive, then from total probability, we have
P (B) = P (B/A)P (A) + P (B/ ¯ A)P ( ¯ A) = 0.01
0.6 + 0.99
0.4 = 0.402
P ( ¯ A/B) = P ( ¯ A
B)/P (B) = 0.99
0.4/0.402 = 0.985
The probability that a woman has positive result and actually carry a baby is
0.985.
The probability that a woman has positive result but does carry a baby is
0.015.
(b). Let
be the event that a healthy woman taking a mammography test and
be
¯ A
A
the event that a woman with breast cancer taking a test, we have
P (A) = 0.999
and
. Let
denote the event that mammography test is positive,
P ( ¯ A) = 0.001
B
then from total probability, we have
P (B) = P (B/A)P (A) + P (B/ ¯ A)P ( ¯ A) = 0.999
0.01 + 0.001
0.99 = 0.01098
P (A/B) = P (A
B)/P (B) = 0.999
0.01/0.01098 = 0.91
The probability that a woman has positive mammography result but is actually
healthy is
(false positive).
0.91
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