## 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? one year ago one year ago

1. experimentX Group Title

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

2. klimenkov Group Title

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 Group Title

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

4. experimentX Group Title

and how long will I wait?

5. klimenkov Group Title

For example, 15 minutes = 1/4 hour.

6. experimentX Group Title

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 Group Title

If you are interested I will give you a solution.

8. klimenkov Group Title

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

9. experimentX Group Title

hold on ... I'll wait.

10. klimenkov Group Title

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 Group Title

m/n??

12. klimenkov Group Title

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 Group Title

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

14. klimenkov Group Title

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

15. experimentX Group Title

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

16. experimentX Group Title

|dw:1350223247938:dw|

17. klimenkov Group Title

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 Group Title

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

19. experimentX Group Title

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

20. klimenkov Group Title

They both can arrive at any time!

21. experimentX Group Title

|dw:1350223462963:dw|

22. klimenkov Group Title

Better to say - you both :)

23. klimenkov Group Title

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

24. klimenkov Group Title

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

25. experimentX Group Title

that is almost zero.

26. klimenkov Group Title

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

27. experimentX Group Title

haha ..

28. klimenkov Group Title

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 Group Title

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

30. klimenkov Group Title

What can you say now?

31. klimenkov Group Title

Did you get it?

32. experimentX Group Title

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

33. klimenkov Group Title

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 Group Title

how to represent waiting time then??

35. klimenkov Group Title

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

36. experimentX Group Title

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

37. klimenkov Group Title

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

38. experimentX Group Title

how to represent this probabilistically?

39. experimentX Group Title

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 Group Title

41. experimentX Group Title

τ is not T right??

42. klimenkov Group Title

Sure!

43. experimentX Group Title

is it (τ /T) ?

44. klimenkov Group Title

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

45. experimentX Group Title

pi t^2/T^2 ??

46. experimentX Group Title

$\piτ^2 \over T^2$

47. klimenkov Group Title

Where did you get pi?

48. experimentX Group Title

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

49. klimenkov Group Title

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

50. klimenkov Group Title

|x−y|<τ

51. experimentX Group Title

yeah, is that correct?

52. klimenkov Group Title

|dw:1350227268062:dw|