Wendy is a randomly chosen member of a large female population, in which 13% are pregnant. Wendy tests positive in a pregnancy test. Pregnancy test correctly identifies pregnancy 91% of the time and correctly identifies non pregnancy 95% of the time. What is the probability that Wendy is pregnant given the positive test result ?

Question

Wendy is a randomly chosen member of a large female population, in which 13% are pregnant. Wendy tests positive in a pregnancy test. Pregnancy test correctly identifies pregnancy 91% of the time and correctly identifies non pregnancy 95% of the time. What is the probability that Wendy is pregnant given the positive test result ?

Answer 1

Why wouldn't it be .91?

Answer 2

To calculate the probability that Wendy is pregnant given the positive test result, we can use Bayes' theorem. Bayes' theorem states that the probability of an event A given that event B has occurred is equal to the probability of event B given that event A has occurred, multiplied by the probability of event A occurring, divided by the probability of event B occurring.

Let's break down the given information:

P(Wendy is pregnant) = 13% = 0.13
P(Wendy is not pregnant) = 1 - P(Wendy is pregnant) = 1 - 0.13 = 0.87

P(Positive test | Wendy is pregnant) = 91% = 0.91
P(Negative test | Wendy is not pregnant) = 95% = 0.95

We are looking for P(Wendy is pregnant | Positive test), which is the probability that Wendy is pregnant given a positive test result.

Now, let's calculate the probability using Bayes' theorem:

P(Wendy is pregnant | Positive test) = (P(Positive test | Wendy is pregnant) * P(Wendy is pregnant)) / P(Positive test)

To find P(Positive test), we need to consider the probabilities of getting a positive test result in both the pregnant and non-pregnant populations:

P(Positive test) = (P(Positive test | Wendy is pregnant) * P(Wendy is pregnant)) + (P(Positive test | Wendy is not pregnant) * P(Wendy is not pregnant))

Substituting the given values into the equation:

P(Positive test) = (0.91 * 0.13) + (0.05 * 0.87)
P(Positive test) = 0.1183 + 0.0435
P(Positive test) = 0.1618

Now, we can substitute this value into the original equation to calculate the probability that Wendy is pregnant given the positive test result:

P(Wendy is pregnant | Positive test) = (0.91 * 0.13) / 0.1618

Calculating this expression:

P(Wendy is pregnant | Positive test) = 0.1183 / 0.1618
P(Wendy is pregnant | Positive test) ≈ 0.729

Therefore, the probability that Wendy is pregnant given the positive test result is approximately 0.729, or 72.9%.