klimenkov
  • 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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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experimentX
  • experimentX
I don't seem to understand this problem ... are they allowed to come at any time?
klimenkov
  • klimenkov
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
  • klimenkov
\([0,T]\) is used to simplify.

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More answers

experimentX
  • experimentX
and how long will I wait?
klimenkov
  • klimenkov
For example, 15 minutes = 1/4 hour.
experimentX
  • experimentX
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
  • klimenkov
If you are interested I will give you a solution.
klimenkov
  • klimenkov
But firstly, I'd like you to solve another problem.
experimentX
  • experimentX
hold on ... I'll wait.
klimenkov
  • klimenkov
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
  • experimentX
m/n??
klimenkov
  • klimenkov
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
  • experimentX
pir^2/4pir^2 = 1/4 ??
klimenkov
  • klimenkov
Yes. Very good! So the probability is the ratio of the areas! The method for my problem is very similar.
experimentX
  • experimentX
all right ... let's get down to this.|dw:1350223223318:dw|
experimentX
  • experimentX
|dw:1350223247938:dw|
klimenkov
  • klimenkov
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
  • klimenkov
No!!! The last one is wrong. Try to get why.
experimentX
  • experimentX
lol ... first one can arrive at any time ...it wouldn't matter.
klimenkov
  • klimenkov
They both can arrive at any time!
experimentX
  • experimentX
|dw:1350223462963:dw|
klimenkov
  • klimenkov
Better to say - you both :)
klimenkov
  • klimenkov
Hm.. Can't get what you do...
klimenkov
  • klimenkov
Lets come back to the shooter. What is the probability to hit in the center of the circle?
experimentX
  • experimentX
that is almost zero.
klimenkov
  • klimenkov
Not almost. It equals zero. Because the area of a point = 0.
experimentX
  • experimentX
haha ..
klimenkov
  • klimenkov
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
  • klimenkov
Oh. Any point of the SQUARE can show when this persons arrived.
klimenkov
  • klimenkov
What can you say now?
klimenkov
  • klimenkov
Did you get it?
experimentX
  • experimentX
no ... just on it. let me try to understand it ...
klimenkov
  • klimenkov
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
  • experimentX
how to represent waiting time then??
klimenkov
  • klimenkov
What about \(|x-y|<\tau\) ?
experimentX
  • experimentX
|dw:1350224492870:dw| can it be independent of x?
klimenkov
  • klimenkov
In my example if \(\tau=21 \) min you will meet your friend. But if \(\tau < 20\) you will not.
experimentX
  • experimentX
how to represent this probabilistically?
experimentX
  • experimentX
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
  • klimenkov
So, what about |x−y|<τ ?
experimentX
  • experimentX
τ is not T right??
klimenkov
  • klimenkov
Sure!
experimentX
  • experimentX
is it (τ /T) ?
klimenkov
  • klimenkov
No. Find all points (x,y) that satisfy the statement of the problem.
experimentX
  • experimentX
pi t^2/T^2 ??
experimentX
  • experimentX
\[ \piτ^2 \over T^2 \]
klimenkov
  • klimenkov
Where did you get pi?
experimentX
  • experimentX
|dw:1350225916366:dw| sorry .. kinda thought of complex number |z| < r
klimenkov
  • klimenkov
The condition that they will meet is \(|x=y|<\tau\). Got it?
klimenkov
  • klimenkov
|x−y|<τ
experimentX
  • experimentX
yeah, is that correct?
klimenkov
  • klimenkov
|dw:1350227268062:dw|

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