Applications of Bayes’ Theorem
Imagine a mother who has two children and you select one child at random. What are the odds that the other child is a boy? Most would say it is 1/2, and this turns out to be correct.
After you selected a child, suppose that nothing changed except that the mother confirms that she has at least one boy. What are the odds now that the other child is a boy? Many may say that it is still 1/2 since nothing really seemed to change. But assuming normal distribution, the probability is no longer 1/2 since we now have new information about the sample space.
As it turns out, Bayes’ Theorem as learned in class is useful in calculating the true probability. In this case, we want P(B | A) where B is the event that the other child is a boy and A is the event that the mother has at least one boy. So P(B | A) = P(A | B)*P(B)/P(A) where P(B) = 1/2, P(A) = 3/4, and P(A | B) = 1. We know P(A) = 3/4 since out of the four possibilities {(boy, boy), (boy, girl), (girl, boy), (girl, girl)}, three possibilities have at least one boy. P(A | B) = 1 because we already know that the mother has at least one boy. P(B) = 1/2 because the other child can be either a boy or a girl with equal chance, assuming we did not know the mother had at least one boy. So P(A | B)*P(B)/P(A) = (1)*(1/2)/(3/4) = 2/3, which is the correct probability for this case.
But Bayes’ Theorem also has critical implications for a variety of real world applications. For example, Bayes is quite relevant in evaluating a real life criminal investigation. In a particular case, two suspects were put to trial for the murder of Aarushi Talwar, a 13-year old girl. A judge gave the suspects heavy sentences and sent them to a life in prison. While many people were relieved that justice was served, some wondered whether the additional testifying witnesses were really at the crime scene or that perhaps further investigation was needed.
The police had gathered those witnesses already and gave them a total of 12 lie detector tests regarding their presence at the scene. Afterwards, all 12 tests indicated that the witnesses were at the crime scene. But what are the odds that they were truly at the crime scene since the lie detector might only have an accuracy of 60%? Also, the probability that an individual witness was at the crime scene is assumed to be ~10% since he is likely an occasional visitor. Using principles from Bayes’ theorem, the article calculated the probability that those three witnesses were present at the crime scene to be ~93.5%.
This probability does seem surprisingly confident especially since the lie detector was assumed to perform not much better than guessing rate. Now that we know that there is at least a 93.5% chance that the three witnesses were at the crime scene, the witnesses probably do not need to be investigated further. Even if this is not quite a 95% probability required in many confidence tests, the lie detector is probably going to be significantly more accurate so that the true probability will exceed 95%. In fact, the witnesses really were not investigated further and the two suspects, Aarushi’s parents, were assuredly sentenced to a life in prison.
Though the other three witnesses were shown to be innocent, a journalist Avirook Sen cannot help but wonder “why would they even unconsciously place themselves there?” Then he remarks “unless that is how it was.” Bayes’ Theorem, combined with various social factors, may also help explain this phenomenon.
Link: http://www.livemint.com/Opinion/9zFN5S6i3XEYoj0OnWeLnJ/A-little-help-from-Bayes.html