# Probability problem

I created this problem based on the following probability riddle here.

You're a king, and you were given two groups of people, and a certain information about them.

• First group has 2 people. One of them is 100% certain to be guilty of a crime. The other one is 50% likely to be guilty and 50% likely to be innocent, and you don't know which one is which.

• Second group is composed by only 1 person. That person is 50% likely to be guilty of a crime and 50% likely to be innocent.

Now you need to imprison either the first group or the second group.

By choosing the first group, what is the probability that you imprison an innocent person? What if you imprison the second group?

My thought is that it wouldn't matter, in both cases is 50%, but according to the video it's not (I suggest checking it). Maybe you guys can shed some light on this.

## migrated from mathematica.stackexchange.comJun 20 '16 at 13:29

This question came from our site for users of Wolfram Mathematica.

• A better question would be locking up one person in the first group, chosen at random from the two in the group. Then the probability of locking up an innocent person is $25\%$. – Henry Jun 20 '16 at 13:45
• @Henry Yes that would be a better question. I really wanted to discuss the video's calculations, a better explanation to them, and if they're correct at all. But thanks for the observation. – Jean Carlos Suárez Marranzini Jun 20 '16 at 14:04
• Given the accepted answer this question is incorrectly formulated. As formulated the correct answer should be 50% as @joriki pointed out. – skyking Jun 22 '16 at 9:34

## 2 Answers

If you choose the first group, one has 0% chance of innocence, and another has 50%. So the average for that one is 25%.
The second group is just one person with a 50% chance, so that one is 50%.

• So why in the video they say it's 67% chance? That's what I'd like to know... I recommend checking it since it's the root of my question. – Jean Carlos Suárez Marranzini Jun 20 '16 at 15:21
• But the question was to imprison either group one or group two. Imprisoning group one includes you imprison both the first and second person - the second person has 50% chance of innocence (which means that you have 50% risk of imprisoning an innocent person). – skyking Jun 22 '16 at 9:39

Your analogy with the video is flawed. The situation you describe is not analogous to the situation described in the video. The video has been discussed at length on this site; see The frog puzzle and also Basic probability : the frog riddle - what are the chances?. Your result that the probability to imprison an innocent person is $50\%$ no matter which group you imprison is correct.