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Omega


Total Posts: 426
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: 1411
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: 1411
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: 47
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: 5011
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: 5011
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: 172
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: 1411
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: 124
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?

mj


Total Posts: 1049
Joined: Jun 2004
 
Posted: 2017-06-15 07:54
I am interpreting the question take one of the three pancakes at random. Flip it randomly. Its top side is burnt. What is the probability the bottom is burnt?

The unconditional probability the top is burnt is 0.5. Since have the pancake sides are burnt.

The unconditional probability that both sides are burnt is 1/3.

The conditional probability is the ratio of these, by definition, which is 2/3.

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