Greetings to all.
I started an MSc that includes a course in statistics. Full disclosure: my bachelor’s had no courses of statics and it is in biology.
So, the professor was trying to explain the Bayes theorem and conditional probability through the following example.
“A friend of yours invites you over. He says he has 2 children. When you go over, a child opens the door for you and it is a boy. What is the probability that the other child is a boy as well.”
The math say the probability the other child is a boy is increased the moment we learn that one of the kids is a boy. Which i cannot wrap my head around, assuming that each birth is a separate event (the fact that a boy was born does not affect the result of the other birth), and the result of each birth can be a boy or a girl with 50/50 chance.
I get that “math says so” but… Could someone please explain? thank you
In: 77
That example is wrong, and the probability is still 0.5.
A much better example of the Bayes theorem is false-positives and false-negatives.Suppose 0.0001% (one millionth) of the population have some disease X. You randomly decide to take a test for disease X.
Suppose the test that you take has a chance of 0.01% (one ten thousandths) of giving a false positive result, and the same probability of having a false negative result. You get your results, and they are positive. Which probability should you assign to having disease X?
Suppose event A is you having the disease, event B is you getting a positive result.
You need to find the probability P(A|B). You know that P(B|A) is 0.9999, and P(B|!A) is 0.0001.Then P(B) is P(B|A) * P(A) + P(B|!A) * P(!A) = 0.9999*0.000001 + 0.0001 * 0.999999= 0.0001009998 .Finally, P(A|B)= P(A) * P(B|A) / P(B) = 0.000001 * 0.9999 / 0.0001009998 = 0.00990002 .
So, your probability of having disease X is still only around 1/100 , despite testing positive with a pretty reliable test.
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