klimenkov Group Title 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? 2 years ago 2 years ago

1. experimentX

I don't seem to understand this problem ... are they allowed to come at any time?

2. 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.

3. klimenkov

$$[0,T]$$ is used to simplify.

4. experimentX

and how long will I wait?

5. klimenkov

For example, 15 minutes = 1/4 hour.

6. 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.

7. klimenkov

If you are interested I will give you a solution.

8. klimenkov

But firstly, I'd like you to solve another problem.

9. experimentX

hold on ... I'll wait.

10. 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)?

11. experimentX

m/n??

12. 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|

13. experimentX

pir^2/4pir^2 = 1/4 ??

14. klimenkov

Yes. Very good! So the probability is the ratio of the areas! The method for my problem is very similar.

15. experimentX

all right ... let's get down to this.|dw:1350223223318:dw|

16. experimentX

|dw:1350223247938:dw|

17. 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 $$(x,y)$$.

18. klimenkov

No!!! The last one is wrong. Try to get why.

19. experimentX

lol ... first one can arrive at any time ...it wouldn't matter.

20. klimenkov

They both can arrive at any time!

21. experimentX

|dw:1350223462963:dw|

22. klimenkov

Better to say - you both :)

23. klimenkov

Hm.. Can't get what you do...

24. klimenkov

Lets come back to the shooter. What is the probability to hit in the center of the circle?

25. experimentX

that is almost zero.

26. klimenkov

Not almost. It equals zero. Because the area of a point = 0.

27. experimentX

haha ..

28. 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.

29. klimenkov

Oh. Any point of the SQUARE can show when this persons arrived.

30. klimenkov

What can you say now?

31. klimenkov

Did you get it?

32. experimentX

no ... just on it. let me try to understand it ...

33. 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|

34. experimentX

how to represent waiting time then??

35. klimenkov

What about $$|x-y|<\tau$$ ?

36. experimentX

|dw:1350224492870:dw| can it be independent of x?

37. klimenkov

In my example if $$\tau=21$$ min you will meet your friend. But if $$\tau < 20$$ you will not.

38. experimentX

how to represent this probabilistically?

39. 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.

40. klimenkov

So, what about |x−y|<τ ?

41. experimentX

τ is not T right??

42. klimenkov

Sure!

43. experimentX

is it (τ /T) ?

44. klimenkov

No. Find all points (x,y) that satisfy the statement of the problem.

45. experimentX

pi t^2/T^2 ??

46. experimentX

$\piτ^2 \over T^2$

47. klimenkov

Where did you get pi?

48. experimentX

|dw:1350225916366:dw| sorry .. kinda thought of complex number |z| < r

49. klimenkov

The condition that they will meet is $$|x=y|<\tau$$. Got it?

50. klimenkov

|x−y|<τ

51. experimentX

yeah, is that correct?

52. klimenkov

|dw:1350227268062:dw|