Pancakes and Bayes' Rule

Nyxie

If we use Baye's Theorem, say A is "we get the pancake burnt on both sides" and B is "we see a burnt side".

P(A|B) is what we're looking for. P(A|B) is the conditional probability of A given that B is true.

With P(B|A) = 1, P(A) = 1/2, P(B) = 3/4
P(A|B) = 2/3

P(B) = 3/4 would be because of the 4 sides of the two pancakes with at least one burnt side, 3 of those sides are burnt.

But this gives the probability where each side of each pancake is considered distinct from the others. Yet we are not looking for the probability that a given side of pancake A is upright. We are looking for either side being upright.

Thus P(B) = 2/3, counting pancake A as one occurence and both sides of the other pancake as 2 other occurences. 2/3 of those occurences will show a burnt side. This gives 3/4 again, so what's going on?

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micromass

That's not what I did. I wrote down the 12 possibilities to make it easier on myself. The sample space is actuallly

(B,B|N,N|N,B)
(N,N|N,B|B,B)
(N,N|B,N|B,B)
(N,N|B,B|N,B)
(B,N|N,N|B,B)
(N,B|N,N|B,B)

And you see from that that the probability is 2/3.

micromass

I'm at loss at how you obtained these probabilities. P(A)=1/3 and P(B)=1/2 to me, and then P(A|B)=2/3, like expected.

No, that's not true. P(B) denotes the probability that we see a burnt side regardless of anything else. Since there are 6 sides with 3 burnt sides, the probability is 1/2. You can't just eliminate the pancake with no burnt sides like that! If you do that then you're conditioning B over some other event, and then you're not calculating P(B).

Karlx

Let's name the pancakes as

[itex]B_{0}[/itex]= Pancake with no burnt sides
[itex]B_{1}[/itex]= Pancake with one burnt side
[itex]B_{2}[/itex]= Pancake with two burnt sides

The probability a priori, for a pancake of being at the top after stacking them is 1/3 for each one. I think there is no doubt about this.
Now, let's take into account that the top side is burnt.
This let us with only [itex]B_{1}[/itex] and [itex]B_{2}[/itex] as being the pancake on the top.
But they don't have the same probability at all.
If the pancake at the top is pancake [itex]B_{2}[/itex], it can appear in 2 possible ways, depending on which of its sides (both burnt) is up.
On the other hand, if the pancake on the top side is pancake [itex]B_{1}[/itex], then it can appear in only 1 way: its burnt side up.
So, the probability that the pancake in the top is [itex]B_{2}[/itex], knowing that the top side is burnt, is clearly 2/3.

kaleidoscope

there are only 3 burnt sides and 2 sides belong to the completely burnt pancake, so the probability is 2/3.