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