This is part 3 of an ongoing series which explores probabilistic thinking through investigating a murder. You can find part 2 here and part 1 here.

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When we last saw Vilila and Wrinje, they were waiting for Glagalbagal to answer a question they had. An old lady in the local area had been murdered, and her niece had been arrested. They wanted to know whether it was likely that the niece had murdered her aunt.

**Vilila**: Hi Glagalbagal. I got you some limes for dinner. But, before that, can you talk to your annoying cousin. He has so many questions. I often feel like smacking him on the head with a rolling pin.

**Glagalbagal**: Of course. He’s just a curious kid. Curiosity is a great trait to have.

**Vilila**: You say curious. I say annoying. Anyways, I’ll leave him to you.

**Glagalbagal**: Wrinje! What was your question?

**Wrinje**: See this article. Glerna, an old lady, was murdered and the police arrested her niece for the murder.

**Glagalbagal**: That’s sad!

**Wrinje**: What’s sad about it?

**Glagalbagal**: Somebody was murdered, right?

**Wrinje**: Oh yeah! I don’t really care about that. I want to figure out whether the niece did the murder.

**Glagalbagal**: Okay. Okay. What does the article say? What evidence does it contain?

**Wrinje**: It doesn’t say anything about evidence.

**Glagalbagal**: So, how can we conclude anything.

**Wrinje**: I was having a discussion with Mum and she first said that the chance that a random person did a murder is very low and so the chance that the niece, Jansu, murdered her aunt is also very low. That seems wrong to me since Jansu is not a random person. She knew the victim.

**Glagalbagal**: That’s interesting. You are right. If we just think of the likelihood that a person we pick off the street did that murder, that is very low. However, since Jansu knows the victim, it seems like she is more likely to have done the murder than, say, you.

**Wrinje**: That does seem right, but I looked up the statistics for our city. Only ten percent of murders are done by people known to the victim. So, doesn’t that make it less likely that Jansu did the murder than a person we pick off the street? That confuses me because intuitively it seems like Jansu is more likely to be the killer than that person.

**Glagabagal**: You are missing something here - the number of people in each of these categories.

**Wrinje**: How does that matter?

**Glagalbagal**: Let us consider a hypothetical village.

**Wrinje**: What’s a hypothetical?

**Glagalbagal**: It just means made up.

**Wrinje**: Why don’t you just say that, then.

**Glagalbagal**: The only reason anybody uses big words is to sound fancy and smart. That’s also why I do it.

**Wrinje**: Adults are weird.

**Glagalbagal**: That’s true. Anyways, getting back to the hypothetical, suppose the village had three people, Glerna, Jansu and Klimpan. Glerna knows Jansu but Klimpan is a loner who doesn’t leave his house. Suppose that Glerna was murdered and we accept that there is a 10% likelihood that it was done by somebody she knew. Who is likely to have done the murder?

**Wrinje**: Klimpan. There is a 90% chance that he did it and a 10% chance that Jansu did it.

**Glagalbagal**: Right. Now, let us introduce another loner who Glerna doesn’t know. His name is Rhayjo. Now, what is the chance that Klimpan did it?

**Wrinje**: 90%, right, because he is not known to Glerna?

**Glagalbagal**: No. Now, that 90% is shared between Klimpan and Rhayjo. So, assuming we know nothing else about them, we can say that there is a 45% chance Klimpan did it and a 45% chance Rhayjo did it. There is still a 10% chance that Jansu did it since she is the only person Glerna knows.

**Wrinje**: That makes sense.

**Glagalbagal**: To make it clearer, let us introduce 90 loners along with Jansu. Now, there is a 90% chance that one of the loners did it which means that there is a 1% chance that a particular loner did it. However, there is still a 10% chance that Jansu did it.

**Wrinje**: That is interesting. Even though there is a 90% chance that a person unknown to Glerna did it, there is only a 1% chance that any one of them did it.

**Glagalbagal**: In this hypothetical. Given the number of people in our city, the chance that a particular stranger did it is almost zero. However, an old lady like Glerna probably knows not more than 50 people. So, there is still a reasonable chance that one of them did it.

**Wrinje**: There are a million people in the city, so the chance that any one of them did it is close to zero. However, even if we take 50 as the number of people Glerna knows, the chance that Jansu did it is a 50th of 10% which is 0.2%. That is also quite small.

**Glagalbagal**: You are right. Given just this information, it is still unlikely that Jansu did the crime even if it is more likely than a stranger. However, we do have some more information available to us. We know two things:

The police arrested her

As far as we know, she was the last person to meet Glerna

Can we say something more given this additional information? Think about it. I’m coming over for dinner with my parents next weekend. We can discuss this then.

Think about Glagalbagal’s question before reading on. You may find drawing representations of the possibilities useful.

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