% of % - Please Help Me Prove My Friend Wrong Here is the situation:  My friend and I are at an impasse.  I believe I'm correct, but he's so damn stubborn he won't believe me.  Also, I'm not the most articulate at explaining things.  Hopefully some of you guys can help me explain this to him in a way he'll understand.  Here is the problem:

A DVD is either at his parent's house or his own.  The probability that it's at his house is 30%.
If the DVD is at his own house, there is a 90% chance it's on the porch, and a 10% chance it's in the living room.
What is the % chance the DVD is on the porch?

My friend says you take 90% of 30% which is 27 and that is the % chance it's on the porch.  Is this correct?  I don't believe so.  

I believe that regardless of where the DVD is, the chance of it being anywhere in his house is still 30% overall.  Location inside his house won't change those odd because the porch and the living room are both part of the house.  If there is a 90% chance it's on the porch, it doesn't change the overall odds of it being in that location.

Now, if you rephrase the question and ask, "The DVD Is either at my parents, my porch, or my living room.  What is the % chance it's on my porch?", the answer is 33%.  If there are three places it could be, then there is 33.333% chance it's on the porch.  Even if it's a 90% chance it's at his house, if there are only three places it can be, it remains the same.

I think the correct way of answering the question is:  There is a 30% chance the DVD is at my house.  If it is at my house, there is a 90% chance it's on my porch.  They are two separate odds and you can't take a percentage of the overall odds since the locations are inside the house. 

Is this correct or am I wrong?  And regardless, please give me your explanation.
Your friend is correct and I'll give you an experiment you can try:

Take two boxes labeled "my house" and "my friend's house" and in the first box put two bags labeled "porch" and "living room". You will also need a marble and a die. Take one die and roll it (we're going to approximate 70% ~ 2/3 here). If it's a 1,2,3 or 4, put your hand in the "my house" box but don't pick a bag yet. If it's a 5 or 6 put it anywhere in the "friend's house" box. If your hand is in the "my house" box then you need to roll again to figure out which bag to put it in. If it's a 1,2,3,4 or 5 (let's say 90% ~ 5/6) put it in the "porch" box, and if it's a 6 put it in the "living room" box. Repeat this exercise until you have a feel for how often it lands in "my porch." Record the trials and see what the odds are. You should be convinced now.

Your reasoning is incorrect here:


  If there is a 90% chance it's on the porch, it doesn't change the overall odds of it being in that location.


No, you changed things. There is not a 90% chance it's on the porch. If - if! - it is in your house, then (and only then!) there is a 90% chance it's on the porch. The magic of the 27% comes from the fact that mathematically - and experimentally, as I hope the above box/bag/marble exercise shows - we know how the "in your house" 30% and the "on the porch" 90% interact. Namely, they interact multiplicatively.

How about this? There is a 30% chance you'll go to New York and a 100% chance you'll go to the Empire State Building if you go to New York (because why else would you go to New York? kidding...). Does this mean there's a 100% chance you'll go to the Empire State Building? Well, since you have to go to NY to go to the ESB, that would mean there's a 100% you'll go to New York - and now we're being contradictory! So this interpretation makes no sense and is never what we mean mathematically or in plain English.


  "The DVD Is either at my parents, my porch, or my living room. What is the % chance it's on my porch?", the answer is 33%.


This is definitely wrong. I'm pretty sure you either have ebola or you don't have ebola. Up to you whether you need to call 911 and get yourself quarantined right away because there's a 50% chance you have ebola. Or perhaps the doctors gave your sick relative 6 months to live, but your relative might live a year or two years or three years or four years or five years, which means there's at least an 86% chance the doctor is wrong.

Now put 10 red m&amp;ms in a bag and 1 blue one. Grab one without looking. Since there's two possibilities, there's a 50% chance it's a blue one, right? So I'll bet you a dollar that it's red and you bet me a dollar that it's blue, and we'll see who's paying for lunch later.
If you have already dictated the odds, you can't then change them and treat the house and places as isolated cases. Your friend is right.