Conditional Probability Using a Table: Pregnancy Test A home pregnancy test was given to women, then the pregnancy was verified through blood tests. The following table shows the home pregnancy test results. Notice how the rows indicate whether the woman was
pregnant or not pregnant, and the columns indicate whether the home pregnancy test
was positive or negative. And we’re asked to find the given conditional probabilities. Looking at our notes, the probability that event B occurs given that event A has happened is represented using this notation here, which is read the probability of B given A has occurred. We can also determine
conditional probability using this formula here. But for our examples, it’ll be easier just to use the table. For our first example, we’re asked to find the probability the test was positive given the woman was pregnant. So because it’s given
the woman was pregnant, we’re only concerned
about the pregnant women, which are in this row here. Notice how 76 women were pregnant. And again, because we’re asked to find the probability the
test was positive given the woman was pregnant, of these 76 pregnant
women notice 71 of them had a positive test. And, therefore, the probability the test was positive given the woman was pregnant is 71 divided by 76, or 71 76ths. Let’s also show the decimal and percentage for this fraction. 71 divided by 76 is approximately 0.9342 to four decimal places. Which is 93.42%. Just keep in mind the fraction is exact, the decimal and percent have been rounded. Next, we’re asked to find the probability the woman was pregnant given the test was positive. So we’re given the test was positive and, therefore, we’re only concerned about this column here that gives the positive results. Notice how there were a total of 75 positive tests. And again, we’re asked
to find the probability the woman was pregnant given the test was positive. So notice of the 75 positive tests, 71 of the women were pregnant. And, therefore, the probability a woman is pregnant given the test is positive is 71 out of 75, or 71 75ths. Which as a decimal to four decimal places is approximately 0.9467. Which is 94.67%. Next, we’re asked to find the probability the test was negative given the woman was pregnant. So again we are given
the woman was pregnant, therefore, we’re only concerned about the row where
the women are pregnant, which is this row here. And again there are 76 pregnant women. And the question is the probability the test was negative given the woman was pregnant. Of the pregnant women, notice five of them had a negative test. The probability the test was negative given the woman was pregnant is five out of 76, or five 76ths. Which to four decimals places is approximately 0.0658. Which is 6.58%. And for the last example,
we’re asked to find the probability the woman was not pregnant given the test was negative. So because we are given
the test was negative, we’re only concerned about the negative test results in this column. Notice how there were a total of 57 negative test results. And notice of the negative test results, 52 of them were not pregnant. And, therefore, the probability the woman was not pregnant given the test was negative is 52 out of 57, or 52 57ths. Which to four decimal places is approximately 0.9123. Which is 91.23%. I hope you found this helpful.