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Omega


Total Posts: 420
Joined: Jul 2006
 
Posted: 2017-05-01 20:01
3 pancake. one is with both sides burned, one is with one side burned, the last one with no sides burned. Combined them in a plate. The top side is burned. what is the probability of the other side is burned?

I either have 1/2 or 3/4...but googling also bring up 2/3

My logic is top side is burnt which is with prob 2/3. Given this info (1/2)/(2/3)=3/4

I suspect I'm wrong but happy to hear where I've gone wrong in this logic

pmrnt


Total Posts: 6
Joined: Mar 2008
 
Posted: 2017-05-01 20:52
the prob is 1/2 = (2/6) / (2/3) = P(bottom burned | top burned) =P(bottom burned & top burned) /P (top burned)
You can draw the decision tree to visualize this: there are only 6 arrangements

ahd


Total Posts: 3
Joined: May 2017
 
Posted: 2017-05-01 23:01
Disagree, but perhaps I misunderstand the assumptions. 3 pancakes, call them A, B and C. A is burned on both sides, B on one side only (call it side 1) and C is not burned. I assume that the stack is equally likely apriori to have A, B or C in any position and that either side is equally likely to be face up or down for a given pancake. So there are 48 equally likely configurations before we start conditioning(6 orderings of A, B, C times 8 ways of choosing up and down sides for a given ordering of 3 pancakes). But we're told that the top is burnt which tells us that the top is one of (A_side1, A_side2, B_side1), bringing us down to 3 choices for top side *2 orderings of remaining two pancakes * 4 ways of choosing up and down sides for a given ordering of those two (middle and bottom) pancakeds = 24 possible configurations. Of these 24, 8 also have a burnt side on the bottom. So the answer is 1/3 just by enumeration. Alternatively, Total prob is P(A is top pancake | top is burnt) P(B is bottom pancake | A is top) P(B is burnt side down) + P(B is top pancake | top is burnt) * P(A is bottom pancake) = 2/3 * 1/2 * 1/2 + 1/3*1/2 = 1/3.

silverside


Total Posts: 1404
Joined: Jun 2004
 
Posted: 2017-05-02 01:24
I get 2/3

Unconditional P(burnt both sides) =P(ABC) +P(ACB) = 1/3

Unconditional P(burnt top side only) =0.5 (P(BAC) +P(BCA)) =1/6

Answer = 1/3 / ( 1/3 + 1/6)=2/3


ahd


Total Posts: 3
Joined: May 2017
 
Posted: 2017-05-02 01:46
Are we solving 1) for the probability that the bottom of the 3-pancake stack is burnt, given that the top of the 3-pancake stack is burnt? Or are we solving 2) for the probability that the underside of the top pancake in the stack is burnt, given that it's top is burnt? I was assuming 1) but I think other people are solving 2). What's the question being asked?

silverside


Total Posts: 1404
Joined: Jun 2004
 
Posted: 2017-05-02 02:02
I read it as "other side of the top pancake" but it is not unambiguous

ahd


Total Posts: 3
Joined: May 2017
 
Posted: 2017-05-02 02:20
I agree that the answer to problem 2 (prob that the underside of the topmost pancake is burnt, given that the top is burnt) is 2/3

AB12358


Total Posts: 45
Joined: Apr 2014
 
Posted: 2017-05-02 04:53
From a previous discussion on this topic elsewhere:

"You have three pancakes. The sides of them are colored as followed because you can't cook.

Goat/Goat
Goat/Car
Car/Car "

chiral3
Founding Member

Total Posts: 4983
Joined: Mar 2004
 
Posted: 2017-05-02 13:22
Nice teaser Monty.

Nonius is Satoshi Nakamoto. 物の哀れ

chiral3
Founding Member

Total Posts: 4983
Joined: Mar 2004
 
Posted: 2017-05-02 14:24
After some thought this isn't really a monty hall. Monty Hall involves introducing new information by opening a door in response to an initial guess, based on 1/3rds prob, which in turn alters the probability from 1/3 to 2/3. In this problem all of the info is known and set a priori.

Nonius is Satoshi Nakamoto. 物の哀れ

billWalker


Total Posts: 170
Joined: Feb 2005
 
Posted: 2017-05-02 16:33
I think it's 1/2.
A = burnt both sides
B = burnt on top


"Plausible regularities may be present but swamped by changes in attendant circumstances." Ole Peters

silverside


Total Posts: 1404
Joined: Jun 2004
 
Posted: 2017-05-02 16:55
@billWalker - similar to my calculations but I get P(B) = 0.5 (by symmetry)

punx120


Total Posts: 2
Joined: Feb 2017
 
Posted: 2017-05-04 03:41
I agree with @silverside, P(B) = 0.5 = 3 side burn / 6 total side.

I also think @pmrnt : P(bottom burned | top burned) =P(bottom burned & top burned) / P (top burned), as it simply rely on conditional probability formula (and not bayes theorem). but got P(top burned) wrong like @billWalker.

So the answer is 2/3.

pmrnt


Total Posts: 6
Joined: Mar 2008
 
Posted: 2017-05-04 17:05
Agreed P(B) = 1/2 and not 2/3.

The earlier assumption was that the pancakes cannot be flipped (2 with top burned and hence 6 arrangements)

lmog


Total Posts: 118
Joined: Mar 2010
 
Posted: 2017-05-13 00:38
Just to be clear, if x=burned and o=not burned, aren't these the only stacks with top burnt?

x x x x x x
o o x x x x

o x o o o x
o x o o x o

x o x o o o
x o o x o o

P(bottomoftoppancake=burned) = 4/6 = 2/3 (EDIT: doh wrote 1/3)

no?
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