klimenkov
Two persons want to meet. They know the place where to meet. The time when one of them come to the place of meeting is equiprobable and lies in the \([0,T]\). Someone of them who comes first will wait \(\tau\) minutes and go away. What is the probability of meeting?
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experimentX
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I don't seem to understand this problem ... are they allowed to come at any time?
klimenkov
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Yes. Any time from 0 to T. For example you want to meet your friend between 1 p.m and 2 p.m. This is the same.
klimenkov
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\([0,T]\) is used to simplify.
experimentX
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and how long will I wait?
klimenkov
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For example, 15 minutes = 1/4 hour.
experimentX
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I'm not good with probability ... but this question seems interesting ... since most of my friends always like about time and distance while waiting.
klimenkov
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If you are interested I will give you a solution.
klimenkov
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But firstly, I'd like you to solve another problem.
experimentX
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hold on ... I'll wait.
klimenkov
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If there are n balls in the very dark room and among them are m white and other n-m are black. What is the probability to take a white ball if you can see nothing and the probabilities for taking any of n balls are the same (equiprobable)?
experimentX
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m/n??
klimenkov
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Yes. Now lets try more complicated what is called a geometric probability. Someone want to hit the zone #1 by shooting from a gun. What is the probability to shot at zone #1 if the shooter always hit in the big circle? |dw:1350222646524:dw|
experimentX
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pir^2/4pir^2 = 1/4 ??
klimenkov
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Yes. Very good! So the probability is the ratio of the areas! The method for my problem is very similar.
experimentX
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all right ... let's get down to this.|dw:1350223223318:dw|
experimentX
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|dw:1350223247938:dw|
klimenkov
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Hint! Let x is the time for the first person for example - for you. And y is for the second one - your friend. So the situation when you will come can be described as the pair \((x,y)\).
klimenkov
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No!!! The last one is wrong. Try to get why.
experimentX
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lol ... first one can arrive at any time ...it wouldn't matter.
klimenkov
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They both can arrive at any time!
experimentX
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|dw:1350223462963:dw|
klimenkov
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Better to say - you both :)
klimenkov
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Hm.. Can't get what you do...
klimenkov
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Lets come back to the shooter. What is the probability to hit in the center of the circle?
experimentX
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that is almost zero.
klimenkov
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Not almost. It equals zero. Because the area of a point = 0.
experimentX
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haha ..
klimenkov
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The same situation is here. The pair of (x,y) describes the situation. So there will be a square.|dw:1350223775014:dw| Any point of this circle can show when this persons arrived. Like the shot in any point of the circle.
klimenkov
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Oh. Any point of the SQUARE can show when this persons arrived.
klimenkov
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What can you say now?
klimenkov
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Did you get it?
experimentX
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no ... just on it. let me try to understand it ...
klimenkov
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For example, you decided to meet you friend near restaurant between 14:00 and 14:30. You can arrive at any time between 14 and 14:30 the same situation for him. Let sign the time when you arrive with x and his time - y. For example you arrives at 14:09 and he at 14:29. This will be the point |dw:1350224152495:dw|
experimentX
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how to represent waiting time then??
klimenkov
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What about \(|x-y|<\tau\) ?
experimentX
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|dw:1350224492870:dw|
can it be independent of x?
klimenkov
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In my example if \(\tau=21 \) min you will meet your friend. But if \(\tau < 20\) you will not.
experimentX
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how to represent this probabilistically?
experimentX
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I thought, if first person comes after time x and waits time t then the probability is
\[ {t \over T-x}\]
but this is not independent of x.
klimenkov
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So, what about |x−y|<τ ?
experimentX
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τ is not T right??
klimenkov
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Sure!
experimentX
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is it (τ /T) ?
klimenkov
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No. Find all points (x,y) that satisfy the statement of the problem.
experimentX
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pi t^2/T^2 ??
experimentX
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\[ \piτ^2 \over T^2 \]
klimenkov
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Where did you get pi?
experimentX
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|dw:1350225916366:dw|
sorry .. kinda thought of complex number |z| < r
klimenkov
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The condition that they will meet is \(|x=y|<\tau\). Got it?
klimenkov
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|x−y|<τ
experimentX
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yeah, is that correct?
klimenkov
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|dw:1350227268062:dw|