Probabilty Riddles

[vmacosta]

[ QUOTE ]
For question 1.
You either picked the right door off the bat or you didn't
Probablity that you picked the right door is 1/3 so the chances that door B is correct is 1/3. Conversly the probability that you picked the wrong door is 2/3 so the chances that the other door is correct is 2/3.

[/ QUOTE ]
Huh? The new information tells you that your choice has 50% chance of being correct so switching does you no good. The answer quoted (switch doors) must be a silly play on words.

[vmacosta]

[ QUOTE ]
Vinegar in the oil, unless I'm missing something, and assuming that the vinegar distributes itself evenly. If there's 10 teaspoons in each jar, then after you put the vinegar in the oil, there's now 11 teaspoons total, with 10/11 oil, 1/11 vinegar. So a teaspoon taken from that jar will contain roughly 91% oil, and 9% vinegar. Which means that the vinegar jar gets some of its vinegar back and 91% of a teaspoon of oil.

I must be missing something, heh.

[/ QUOTE ]
Actually, by your logic, they both get the same amount of impurity (10-1+1/11=100/11 for vinegar and 10-10/11=100/11 for oil). Is the answer based on lower density of oil or do we assume uniform mixing?

[uphigh_downlow]

equal amounts in each, if you start with equal volumes in the jars

Xo and Xv to start X is the # spoonfullss in each

Xo + 1v and (X-1)v after the first exchange

(Xo + 1v){1- (1/X+1)} and (X-1)v + (Xo +1v)/X+1
after second exchange

Turns out that there is X/(X+1) of one in the other, both ways

[LetYouDown]

Yeah, I got the concept but didn't even think about it. I vote for same amount.

[Thythe]

[ QUOTE ]
[ QUOTE ]
For question 1.
You either picked the right door off the bat or you didn't
Probablity that you picked the right door is 1/3 so the chances that door B is correct is 1/3. Conversly the probability that you picked the wrong door is 2/3 so the chances that the other door is correct is 2/3.

[/ QUOTE ]
Huh? The new information tells you that your choice has 50% chance of being correct so switching does you no good. The answer quoted (switch doors) must be a silly play on words.

[/ QUOTE ]

No, switching doors is indeed correct. I know I'm just restating the above, but think of it this way. If you pick the wrong door, the host shows you the other wrong door. You then switch and get the correct door. What is the probability of picking the wrong door originally? 2/3. Now if you pick the right door originally, the host shows you randomly one of the two wrong doors, and then you unfortunately switch to the other wrong one. What is the probability of picking the right door originally? 1/3. So it is indeed correct to switch and you will win with 2/3 probability.

[sirtimo]

I always found it easier to think of that question using 100 doors-

You pick one-
The host opens 98 empty ones-
Should you keep the one you originaly picked or the remaining unknown door?

Much easier to grasp the concept that way.

regards,
Tim

[gaming_mouse]

[ QUOTE ]
equal amounts in each, if you start with equal volumes in the jars

Xo and Xv to start X is the # spoonfullss in each

Xo + 1v and (X-1)v after the first exchange

(Xo + 1v){1- (1/X+1)} and (X-1)v + (Xo +1v)/X+1
after second exchange

Turns out that there is X/(X+1) of one in the other, both ways

[/ QUOTE ]

Your answer is correct, but I believe this calculation is assuming that they are getting evenly mixed, which you cannot assume. That's why the question uses oil and vinegar. In fact, you cannot know how much of each will be in the other -- there might be 0, and there might be a full spoonful. But even so, you CAN always know that the amount of contamination will be equal. There's a simple logical argument, which requires no algebra, and is the reason I like the puzzle.

gm

[RocketManJames]

When I first heard this problem a couple years ago, I ended up reasoning it out using jars of marbles. One jar had N red marbles, and another had N blue marbles.

Using this approach, it was pretty easy to see why the contamination was equal.

-RMJ

[gaming_mouse]

[ QUOTE ]
When I first heard this problem a couple years ago, I ended up reasoning it out using jars of marbles. One jar had N red marbles, and another had N blue marbles.

Using this approach, it was pretty easy to see why the contamination was equal.

-RMJ

[/ QUOTE ]

Yes, that works. Or just notice that the amount of oil being transferred over to the vinegar side is exactly the same as the amount of vinegar which remains in the oil side. To put it mathematcially:

(vinegar being returned to vinegar) + (oil moving to vinegar) = 1 spoonful
(vinegar being returned to vinegar) + (vinegar left over in oil) = 1 spoonful

The result follows.

[chaosuk]

It is perhaps simpler and key just to acknowledge that both jars finished with the same volume of liquid they started with.Since the liquids are conserved between the jars, whatever one jar lost the other jar gained. Whatever they lost must have been replaced by an equal volume from the other jar - the jars simply exchange a volume of eachothers liquid which muist be the same since, they both end up with the the same volume they started with.

EG If the vinegar jar has 75% vinegar left then it must have 25% oil. So the oil jar must have 75oil, 25% vinegar.