Asha has a child. What’s the chance that it’s a girl? Simple, you think: 1/2, or 50-50.

Asad has two children. What’s the chance that one is a girl? Simple, you think: 1/2 again, right? Wrong. There are four ways to have two kids: girl-girl, girl-boy, boy-girl and boy-boy. There’s a girl in three of those. Thus the chance that one of Asad’s kids is a girl is 3/4.

With me so far? Okay, Archana also has two kids. She tells you that one is a girl. What’s the chance that the other is a girl too? Simple, you think: the other child is either a boy or a girl, so, well, it’s 50-50 again … but wait a minute. If we know one of the two is a girl, then we have three possible cases: girl-girl, girl-boy and boy-girl. In only one of those is the second child a girl. Thus the chance that Archana’s other child is a girl is 1/3.

The question is, effectively, whether both Archana’s kids are girls. If we know nothing about them, the chance of that is 1/4 (girl-girl, out of four possibilities). But we do know something about them—that one is a girl. That changes the probability to 1/3.

There’s an almost counter-intuitive idea at work here: a given bit of knowledge—one of Archana’s two kids is a girl—affects our calculations of chance. We’re skating here in the realm of conditional probabilities and a mathematical rule attributed to Thomas Bayes. Bayes’ law tells us how the probability of something happening changes when we get new evidence about it.

I use evidence advisedly, as perhaps you’ll soon see.

Bayes’ law applies in many areas—drug testing, cancer treatments, blocking email spam. It also helps examine situations and make important decisions. For example, say you’ve smoked all your life. You know that makes you susceptible to cancer. Your doctor finds a growth in your throat. He sends a sample for a biopsy, telling you the test is accurate 75% of the time. The result is positive. What is the probability that you actually do have cancer?

Bayes will calculate that. The answer might even surprise you.

Take another example. After a terrible crime, the police find two suspects, charge them and put them on trial. They are found guilty. The judge gives them heavy sentences and they disappear into jail. Everyone is mighty relieved because the two monsters are where they belong.

Only… people start asking questions. Three other men may have been at the scene when the crime happened. The police actually questioned them. Not only that, the police also gave them lie-detector tests—12 times. All three made statements that placed all three at the scene: in effect, this made each man both witness and suspect. All 12 tests were positive.

What is the probability that these three were really at the crime scene?

Bayes can help answer that. We start by making some reasonable assumptions. First, what’s the chance that any one of the three was at the scene when the crime was committed? Perhaps they were occasional visitors, dropping by about once every 10 days. So, call that chance 10%, or 0.1. Next, what’s the probability the lie-detector test is accurate? Say it’s 60%, meaning 0.6. Finally, remember that the test was administered 12 times.

Using Bayes’s law with those assumptions, we can figure out how likely it was that the men were there. I’ll spare you the calculations, but here’s the result: 93.5%.

Yes, that’s right. With these numbers, you can be 93.5% sure that the men were there (with less conservative numbers, better than 99%). You can be close to certain, really, that they were in the house when Aarushi was murdered.

Yes, the same 13-year-old Aarushi. Yes, there were three such men. Yes, the authorities ran three different tests—polygraph (lie-detector), brain-mapping and narco-analysis, whose different results behavioral scientists consider together—on the men, a total of 12 altogether. Yes, while taking these tests, each man placed himself in the Talwars’ home when Aarushi was murdered.

Would you call a 93.5% chance a reasonable doubt about their presence there, not worth investigating further? Because these men were never investigated further. As you probably know, Aarushi’s parents were found guilty of her murder, and are serving a life sentence.

Deep in his new book on Aarushi Talwar’s case, journalist Avirook Sen tells this story of the three men, quoting extensively from the results of the tests. With a little help from Bayes, we’re left with the same simple question he asks: “If they were innocent, why would they even unconsciously place themselves there?"

His next sentence: “Unless that is how it was."

Think of that, with a little help from Bayes.

PS: Thanks to Raj Aradhyula and Amritanshu Prasad for their crystal-clear explanations of Bayes.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences.

Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza--

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