[Guest]

You enter the following tunnel system at the starting point (marked "S"). If you exit through the tunnel marked "W", you win a fabulous prize. If you exit through a tunnel marked "L", you lose and get nothing.

At each intersection, you must make a choice which to go. You may only go East, West, or South; never North. You may not backtrack. So, at each intersection you flip a coin to decide which direction to proceed.

What is the probability that you will win the fabulous prize?

Note: There is ONE path where using the above rules can get you stuck, i.e. you cannot make a valid move. For this path ONLY, assume backtracking is ok.

[Guest]

I think there is more than one path that gets you stuck, but only one location where that may happen where backtracking will let you win.

25% chance of getting there.

[Guest]

I think there is more than one path that gets you stuck, but only one location where that may happen where backtracking will let you win.

You are right, I should have worded that better. There is one LOCATION where you can get stuck, although multiple paths to get there.

[Guest]

I think there is more than one location where you either have to go north or backtrack.

In the image is four locations where this is the case. Maybe I missed something?

[Guest]

I think there is more than one location where you either have to go north or backtrack.

In the image is four locations where this is the case. Maybe I missed something?

Hmm I guess you are right... And I thought I had thought this through better

Ok, so let me correct things a little:

The ONLY time you can backtrack is if you cannot make a valid move given the previous rules.

I should have someone check my puzzles before i post them

[Guest]

25% also. Although I thought it was wrong when I first saw it.

There are 9 possible paths that lead to L's and 6 possible paths that lead to W. BUT, there are only 3 paths that branch into the 6 winners, once your on one of those three, you'll reach W.

So 3/12 =25%

[Guest]

I think it may suffice to say you can't move north at an intersection. If you are on a one-way tunnel you are able to move north.

Edited February 12, 2009 by Llam4

[Guest]

25% also. Although I thought it was wrong when I first saw it.

There are 9 possible paths that lead to L's and 6 possible paths that lead to W. BUT, there are only 3 paths that branch into the 6 winners, once your on one of those three, you'll reach W.

So 3/12 =25%

I'm not sure I follow your logic... could you explain a bit more?

[Guest]

I'm not sure I follow your logic... could you explain a bit more?

Looking at my logic... im not too sure if I understand it either, lol. I'll take a look at it agian.

15 possible paths, 9 lead to L, 6 lead to W. BUT! there are 3 routes that reach a point where every possibele path past them leads to the W.

the problem with my answer before was i neglected to find how many routes lead to a similar point, which always leads to L no matter which path is taken after that point. I hope i didnt create more confusion.

ALSO, I never bothered with the probablity of the coin tosses. maybe Im way off.

one more thing: Llam4, if you can move north at all, that makes for an infinite number of possible paths, making the probability essentially 0% that you would make it to W. i think...

[Guest]

25% also. Although I thought it was wrong when I first saw it.

There are 9 possible paths that lead to L's and 6 possible paths that lead to W. BUT, there are only 3 paths that branch into the 6 winners, once your on one of those three, you'll reach W.

So 3/12 =25%

You calculate probability with total paths leading to destination/total paths available.

That means that you take the six winners, not three winners.

Thats probably why my answer is different:

There are fifteen possible routes, and six routes that lead to the prize (9 to L, 6 to W)

So the probability of getting the right spot is 6/15=40%

[Guest]

you can get the the first L, 2 ways, second L 4 ways, third L 5 ways, and the W 11 ways.

11/22 = 50% probability

[Guest]

you can get the the first L, 2 ways, second L 4 ways, third L 5 ways, and the W 11 ways.

11/22 = 50% probability

Ahh, but not all those ways are equally likely!

[Guest]

6.25%. I reduced all the possibilities of loosing the game. When you begin, you have -50% to win, then in the next turn (make some maths) you have - 32,5%, then if you get the right path again (both of them will do) you wil have -6.25% so (-50) + (-32.5) + (-6.25) = -93.75. So thats the chance of L and 6.25% are the chances for W.

[Guest]

Looking at my logic... im not too sure if I understand it either, lol. I'll take a look at it agian.

15 possible paths, 9 lead to L, 6 lead to W. BUT! there are 3 routes that reach a point where every possibele path past them leads to the W.

the problem with my answer before was i neglected to find how many routes lead to a similar point, which always leads to L no matter which path is taken after that point. I hope i didnt create more confusion.

ALSO, I never bothered with the probablity of the coin tosses. maybe Im way off.

one more thing: Llam4, if you can move north at all, that makes for an infinite number of possible paths, making the probability essentially 0% that you would make it to W. i think...

I tried a different method, and was surprised to get the same result, 25%

I labeled the image below with 6 meaningful decision points.

At A, you have 50% chance of losing and 50% left to consider: L + 0.50

At B, you have 50% of that 50%, or 25% chance of going South. At C, you have 50% of that 25% of going South and losing, so L + 0.125

At F, you have 50% of the remaining 12.5% of going S (L) or E (W), so L + 0.0625 and W + 0.0625

At D, you have 50% of 25% of going E and winning, so W + 0.125

At E, you are carrying 12.5% chance and you have 50/50 of W/L, so L + 0.0625 and W + 0.0625

Losing chances are 0.50 + 0.125 + 0.0625 + 0.0625 = 0.75

Winning chances are 0.125 + 0.0625 + 0.0625 = 0.25

QED?

[Prof. Templeton]

I think it may suffice to say you can't move north at an intersection. If you are on a one-way tunnel you are able to move north.

It looks like that would increase the number of W paths

7 or more.

Not to mention the L paths.

The coin tosses don't matter.

[Guest]

I put all the possible paths that I could find that end at W. Then I calculated how many decisions for each path, multiplying by 50% each time. All paths had either 4 or 5 decisions before they got to a point where winning was inevitable. When I summed all of those paths together I came up with:

34.375% chance to win

[Guest]

It looks like that would increase the number of W paths

The coin tosses don't matter.

7 or more.

Not to mention the L paths.

Never North! The OP said Never North!

[Guest]

10.9375

i looked at all the possible ways of getting to W, and each time you had to make a choice, for every choice cuts the probability in half....added them all up and got that number...chances are i made a mistake in the math/counting the choices, but i think the theory is sound for solving it

did it the same way as you, but my max number of choices was up to 8?, any thoughts

Edited February 12, 2009 by michael531

[Guest]

Need clarification on the north thing.

Edited February 12, 2009 by Llam4

[Guest]

Need clarification on the north thing.

IMO, the rule can not go north only applies to intersections, intersections being a spot where you have to make a choice in which direction to travel, if there is no choice to be made then you are not at an intersection and can go north.....In other words there are no dead ends, you will eventually get to a L or the W, since you cant back track

Edited February 12, 2009 by michael531

[Guest]

did it the same way as you, but my max number of choices was up to 8?, any thoughts

some of my paths did not get all the way to W in 5 choices, but they would inevitably get there regardless of what choices were made after that point.

[Guest]

Not true. You can't go north at an intersection, meaning the only time you'd be able to travel north is if you're on one of the paths that requires you to go north to continue to the next intersection.

But you can still end up going in circles. The requirement of not going north must stand or you cannot get a finite number of paths to the end. You can go around a loop as many times as you want before exiting.

For example:

Granted, there is still a way to get a solution even with looped paths included, but the answer will not be the same as for the original puzzle. Thus the rule stands: no North, ever.

Edit: Still waiting for a right answer

Edited February 12, 2009 by dwilly

[Guest]

IMO, the rule can not go north only applies to intersections, intersections being a spot where you have to make a choice in which direction to travel, if there is no choice to be made then you are not at an intersection and can go north.....In other words there are no dead ends, you will eventually get to a L or the W, since you cant back track

It seems really clear from the OP, never go North, backtrack if you have to. Period.

[Guest]

Thus the rule stands: no North, ever.

My method yields 25%

Three paths lead to W:

Path 1:

East-East-East

Path 2:

East - East - South -South

Path 3:

East - South - East - East

Paths 2 & 3 have probability 1/16 each, Path 1 is 1/8.

Sum is 1/4

[Guest]

It appears to me the probablitity would be much lower than 25%. Assuming that a flip that makes you head North is an automatic L (you cannot go North), then there are only 5 paths that lead to the W. The probability of taking anyone path as opposed to another would be equalt to 1/2^n, where n is the number of intersections between S and W. By adding the probabilities of each individual paths, that should give the probability to get to W. This may be wrong but I came up with 4.29% probability of making it to the W.

[Guest]

Still waiting for a right answer

Ok I made a bit of a mistake here:

neida and DanCDow got the right answer quite awhile ago, but neither gave a satisfactory justification.

voltage, you are correct, and your logic is good.

raschaller, if you cannot go North, then you would never flip the coin to choose between North and another direction. It is not an automatic L, it is simply a way to constrain your movement through the maze.

My apologies for the confusion over backtracking and heading north. I need to try to think these things out a little better to avoid confusion