# Pancakes and Bayes' Rule

by richardwander
Tags: bayes, pancakes, rule
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P: 15,673
 Quote by Nyxie micromass - you can't count the unburnt pancake because we already know that it is not the one on top. Unburnt pancake has zero prob. that it could usurp the fully burnt pancake's place, so only the two with burnt sides provide the sample space.
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.
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 P: 75 Let's name the pancakes as $B_{0}$= Pancake with no burnt sides $B_{1}$= Pancake with one burnt side $B_{2}$= 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 $B_{1}$ and $B_{2}$ 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 $B_{2}$, 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 $B_{1}$, then it can appear in only 1 way: its burnt side up. So, the probability that the pancake in the top is $B_{2}$, knowing that the top side is burnt, is clearly 2/3.