Author Topic: Final Decision (Read 7358 times)

dedalos · « **Reply #75 on:** November 29, 2004, 03:32:25 PM »

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Originally posted by midnight Target
The left one is already selected... drop it now, then select and pickle off the right one. Duh!

ccvi · « **Reply #76 on:** November 29, 2004, 05:31:46 PM »

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Originally posted by MANDO
ccvi, by doing so you are going to "ruin" the "good luck" of the pilot for the described problem. That luck is mainly based on the selection of the lost bomb based on two other random selections. If we look for a pure random combinations where the lost bomb is red, the real bomb is green and the lever marks a present bomb, then there is no more "good luck" initially present.

The original case indicates that the pilot see a red dot falling. If you repeat that case 1000 times, he will see the red dot 1000 times because it is part of the description. He may ask himself, ok, what if the ground crew painted the real bomb red by mistake? But what he see is red, not green.

The selection in you solution attempt is not "by luck" but "by design", whereas in the described scenario the selected bombs drops by "luck", not "design".

MANDO · « **Reply #77 on:** November 29, 2004, 05:53:46 PM »

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Originally posted by ccvi
The selection in you solution attempt is not "by luck" but "by design", whereas in the described scenario the selected bombs drops by "luck", not "design".

How do you know that? You know that the lost bomb is red, you dont know that the lost bomb may be green. You may consider that only training bombs can be lost by its engeenering or design. The case describes a single mechanical faillure that affected a training bomb.

If you want to look at the scenario considering that there is no "pre-designed good luck" (the lost bomb is red and always will be red), then you should consider also the following:

1 - The ground crew commit mistakes. In fact, in the described case mistook 2 of 3 (2 reds and 1 green instead of 3 greens). You may suspect that the ground crew doesnt know to differentiate between real and training bombs. So, they may repeat mistakes loading bombs (all training, for example, 1 training plus two real, all real, may be they even forget to load some pilons ...).

2 - The plane may have a mechanical faillure in any bomb and/or pilon, so, more than 1 mechanical faillure are also possible. So, due ground crew mistakes, even losing 2 bombs does not guarantee that the third is green.

3 - The ground crew may load all real bombs also.

4 - The plane may have also NO mechanical failures.

ccvi · « **Reply #78 on:** November 30, 2004, 02:32:24 PM »

You can add all those four cases to your simulation. If you're doing it correctly, the result will not change.
All cases of "might have happend but did not happen" (possible, but do not match the story) need to be treated in the same way: Return() before evaluation.
This is true for those four cases as well as for an unlucky drop of the real bomb or a drop of the bomb that was selected.

MANDO · « **Reply #79 on:** November 30, 2004, 03:14:59 PM »

ccvi, I can add those factors, but they are not related to the initial problem.

You saw a red bomb falling, and you conclude that green bombs may fall also. This is too much of an asumption before having a single case where the green bomb is lost. A real (green)bomb is a different object than a training (red) bomb. Is like saying, I saw a red bomb falling once, so, next time a wing may fall also or even the pilot. Losing a green bomb is not a "possible case" without a single real intance, else, losing anything (not only real bombs) should be possible also.

In the entire history of the events, 1 case only, only one red bomb was lost from a pilon. All the pilons are the same type, so you should assume that the problem was in the red bombs. The only factor that we may add without breaking the iniital case is considering that the second red bomb may also be lost. For these cases the pilot will be success all the times.

BlauK · « **Reply #80 on:** November 30, 2004, 04:09:49 PM »

ccvi,
howabout considering it this way:

The possibility of having the real one originally selected is 1/3 and either of the training bombs (=tb) selected is 2/3.

Then one tb is lost from unselected position.....

If real bomb is selected, losing tb1 is 1/6 and losing tb2 is also 1/6 of the chances.

If tb1 is selected, the lost one was tb2, 1/3 possibility.

If tb2 was selected, the lost one was tb1, 1/3 possibility.

So we end up in a situation with only 2 bombs left and the chances are 1/3 that the real one is selected, 2/3 that a tb is selected..... should you switch?

Where is the fault in the above logic with these given conditions?

It is all about do you switch or not!!! It is not about which of the 2 should you select, since one of them is already selected.

-------------

What is interesting is that one can actually still make 3 choices: decide to switch, decide not to switch or not to decide at all

.... in 2/3 of these cases the selection is not switched. It looks like the odds are against going for better chances (=switching)

ccvi · « **Reply #81 on:** November 30, 2004, 05:09:21 PM »

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Originally posted by MANDO
In the entire history of the events, 1 case only, only one red bomb was lost from a pilon.

You cannot do statistics with a single event.

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Originally posted by BlauK
Where is the fault in the above logic with these given conditions?

The fault is that the solution does not match the initial description of this scenario. "He was lucky, it was one of the two training bombs." If only duds could drop, he wouldn't have been lucky, because no matter how often he was in that situation always duds would have dropped.

Re-check the original post closely:
- "ground crew loaded 2 training bombs (red) and only 1 real bomb (dark green)" - no order given, therefore assumed to be random
- "This lever was at the lower position, so left pilon was selected."
- "the center pilon led switched off, by some mechanical problem that bomb was released"

These are fixed. With a random order of the bombs there's a probability of 1/3 that the real one was dropped and he could have rtb'ed.

But "He was lucky, it was one of the two training bombs."

There's a probability of 1/3 that the real one is on the left wing, and a probability of 1/3 that the real one is on the right wing. With the center bomb gone it's equally likely that the real one is on the left or right wing.

To get 2/3 for the right wing you either need a special the aircraft that whenever a real bomb is loaded to th
e center position moves it to the right wing, or a wizzard that whenever the real bomb was at the center and about to drop catches it and drops the dud on the right wing instead. Neither of these exists.

MANDO · « **Reply #82 on:** November 30, 2004, 05:55:50 PM »

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Originally posted by ccvi
There's a probability of 1/3 that the real one is on the left wing, and a probability of 1/3 that the real one is on the right wing. With the center bomb gone it's equally likely that the real one is on the left or right wing.

Wrong asumption. The real bomb was as safe as any other part of the plane that didnt fall, the problem was on a training one. You only can assume that losing both training bombs is also possible.

You decided to put the "blame" on the pilon instead on the red bomb. There are three pilons, all are the same, there are two training bombs, a combination of a pilon/training bomb failed. Suspecting on the training bombs: 1 failed, 1 worked. Suspecting on the pilons: 1 failed, 2 worked. It is more probable that the problem is on the training bomb, not the pilon. In any case, you cannot suspect on the real bomb itself, it remained and was only one: 0 failed, 1 worked.

If you want the a definitive excuse for the expresion "he was lucky" then consider the following:

When the pilot saw the "red" led (indicator of bomb presence) switching off, he may suspect also on the led itself. When the pilot saw the red bomb falling, he confirmed that the led was OK. That gave him the chance of getting 66% of success. Being the faillure on the led instead of on the bomb, the success chance would be 33%.

BlauK · « **Reply #83 on:** December 01, 2004, 02:53:40 AM »

ccvi,

it is not statistics... it is probabilities

ccvi · « **Reply #84 on:** December 01, 2004, 04:54:15 PM »

The 2/3 for switching would be true if no red bomb and no pylon a red bomb is attached to and no screws that hold a pylon that red bomb is attached to ... had ever failed and wont in future.

Apart from that beeing pretty far fetched because such failure free systems do not exist in reality, which point in your initial story describes that kind of warped reality? Please provide appropriate qutotes from the initial post.

BlauK · « **Reply #85 on:** December 02, 2004, 03:29:08 AM »

Sorry ccvi, I missed what was your point.

Are you trying to say that The Monty Hall example would not work if this was the very first show and he had never before opened a door with a goat for anyone? What does it matter if the door could get stuck, or the handle would fall off when opening, etc? The case states that the door IS already open... or that one bomb is already lost.

It is about probability of switching the selection in this given situation.... "someone had three doors of which one was selected and saw that one of the unselected doors had a goat" ..... or "you had three bombs of which one was selected and you saw that une unselected pylon had a training bomb".

Is it not irrelevant for the actual question of switching, how you get the information of outruling one unselected option??? A man opened the door (that door is missing form your further options) or you saw one bomb drop by itself (it also now misses).

The point is:
You are in this situation now... should you switch the selection? What are the probabilities for getting a live bomb by switching and by not switching? Forget the screws and pylons.

ccvi · « **Reply #86 on:** December 02, 2004, 04:03:44 PM »

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Originally posted by BlauK
Is it not irrelevant for the actual question of switching, how you get the information of outruling one unselected option???

It is relevant. Only if both the probability of losing a real bomb and the probability of losing a bomb that is selected are 0 you'll get the real one by switching in 2/3 of the attempts. Both these conditions are not described in the initial story.

The way to get into such a situation is important. Imagine a slight modifed version of mandos warped reality. Keep the probability of 0 for failures of bombs that are selected. Because of cheap mass-production during war time real bombs tend to fail a lot more than old training bombs from pre-war times. Now imagine a very very lucky pilot that lost a training bomb in exactly the situation mando described. Do you think he has to switch, just because he's in that situation? The only thing that saves his bomb is that the selected one never drops, even if it is real.

dedalos · « **Reply #87 on:** December 02, 2004, 04:24:12 PM »

http://astro.uchicago.edu/rranch/vkashyap/Misc/mh.html

MANDO · « **Reply #88 on:** December 02, 2004, 04:44:52 PM »

ccvi, you cannot make asumptions, you should work only with the data presented in the case, and there is a single case. Based on the described case, you know that red bombs fail, green bombs dont. Probability of losing a red bomb that is selected by the lever is irrelevant. If you lost the selected red bomb, you will need to switch the lever also. Switching the lever will increase always your chances from 33% to 66% or from 0% to 50%.

BlauK · « **Reply #89 on:** December 03, 2004, 05:56:31 AM »

ccvi,
it is exactly like you described. This is that exact case. You are there.. you HAVE LOST that one red bomb. What will you do NOW?

Mando,
If the selected bomb is lost, then it is a whole different story and situation. It is not about switching anymore since nothing is selcted. It is then about selection one of the 2 remaining bombs and the chances are then 50:50