% of % - Please Help Me Prove My Friend Wrong

Solution 1:

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&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.

Solution 2:

The DVD is

• at his parents' house with a probability of $70\,\%$ ($=100\,\%-30\,\%$)
• in the porch of his house with $27\,\%$ ($=90\,\%$ of $30\,\%$)
• in the living room of his house with $3\,\%$ ($=10\,\%$ of $30\,\%$)

Check: $70\,\% + 27\,\%+3\,\%=100\,\%$.

Solution 3:

Here are two ways to think about probability, which I often find helpful.

The Frequentist Interpretation

In the frequentist interpretation of probability, you have a large number of situations set up the same way, and the probability of something being true tells you the fraction of those situations in which the thing is true.

Suppose there are a million parallel universes, each with its own version of you, your friend, your houses, and the DVD. Then the statements your friend makes have the following consequences:

The probability that it's at his house is 30%.

This means that in 30% of the universes, that's 300,000 of them, the DVD is at your friend's house.

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.

Let me take this piece by piece because it's the most important part:

• If the DVD is at his own house,

  This means that you have to only consider the universes where the DVD is at your friend's house. There are 300,000 of these. You have to forget about the rest of the universes for now.

• there is a 90% chance it's on the porch

  In 90% of the universes, the DVD is on the porch. But we're pretending there are only 300,000 universes. So in 90% of those, or 270,000 universes, the DVD is on the porch.

• and a 10% chance it's in the living room.

  Again, we're pretending there are 300,000 universes in all. In 10% of those, or 30,000, the DVD is in the living room.

Okay, those are all the statements, so time to stop pretending and go back to considering all million universes. We have the following totals:

• 30,000 universes where the DVD is in the living room at your friend's house
• 270,000 universes where the DVD is on the porch at your friend's house
• 700,000 universes where the DVD is at his parents' house

To find the probability of the DVD being on the porch, you take the number of universes where the DVD is on the porch and divide it by the total number of universes.

$$P(\text{DVD on porch}) = \frac{\text{# of universes where it's on the porch}}{\text{total # of universes}} = \frac{270\,000}{1\,000\,000} = 27%$$

And similarly for the other cases:

$$\begin{align} P(\text{DVD in living room}) \\ &= \frac{\text{# of universes where it's in the living room}}{\text{total # of universes}} \\ &= \frac{30\,000}{1\,000\,000} \\ &= 3\% \end{align}$$ and $$\begin{align} P(\text{DVD at parents' house}) &= \frac{\text{# of universes where it's at his parents' house}}{\text{total # of universes}} \\ &= \frac{700\,000}{1\,000\,000} \\ &= 70\% \end{align}$$

You can also combine cases: $$\begin{align} P(\text{DVD at friend's house}) &= \frac{\text{# of universes where it's at your friend's house}}{\text{total # of universes}} \\ &= \frac{30\,000 + 270\,000}{1\,000\,000} \\ &= 30\% \end{align}$$ which was an assumption from the start, so of course that has to be the result - but it's good to see that the math works out.

If you tried to say

If there are three places it could be, then there is 33.333% chance it's on the porch.

then you would be claiming that in 333,333 of the universes, the DVD is on the porch. That clearly conflicts with what we calculated, that the DVD is on the porch in only 270,000 universes! So if the earlier statements about probability (30% that it's at his house, etc.) are correct, this latest statement cannot also be correct. You can't assume that the probability of something being true is $1/N$ just because there are $N$ possibilities! Not unless you know, somehow, that all the possibilities are equally likely. (In fact, I didn't say this before, but when I invented the million universes, I assumed that each universe is equally likely. That's important.)

The Bayesian Interpretation

In the Bayesian interpretation, probability is a reflection of how much you know or don't know about some system. It seems quite similar to the frequentist interpretation, at first, but it works a little differently when you start talking about conditional probabilities ("if X then the probability of Y is Z").

To explain this, let me go back to the universes. We started with a million of them. Then you said

The probability that it's at his house is 30%

which means that 300,000 universes have the DVD at your friend's house. OK, that much is the same as the frequentist interpretation.

Then move along to the next statement, and again I'll take it piece by piece:

• If the DVD is at his own house,

  OK, now we're saying you have determined that the DVD is at your friend's house. This is where the Bayesian intepretation differs from the frequentist interpretation: for the rest of this statement, we'll say you know that the DVD is at your friend's house. So you can literally throw out the 700,000 universes where that is not the case. They don't exist anymore.

  Like I said, it's a pretty subtle difference.

  One consequence of this, by the way, is that the probability of the DVD being at your friend's house is now 100%. Or to be more precise, when you found out that the DVD was at your friend's house, you updated the probability of the DVD being at your friend's house from 30% (that's 300,000/1,000,000) to 100% (that's 300,000/300,000).

• there is a 90% chance it's on the porch

  In 90% of the universes, the DVD is on the porch. There are 300,000 universes, so in 90% of those, or 270,000 universes, the DVD is on the porch.

• and a 10% chance it's in the living room.

  In 10% of 300,000 universes, or 30,000 of them, the DVD is in the living room.

If you now back up to the point where you didn't know the DVD was at your friend's house, you'll see that the probabilities wind up being the same as in the frequentist interpretation. (That's true in general. These interpretations are just different ways to think about probability, but they produce the same results.)

There are mathematical procedures for "reversing" an assumption that you made, but I won't get into that level of detail. The point is just that Bayesian probability is a reflection of your knowledge of the system, and that you update probabilities as you learn more about it.