A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

Kainui

  • one year ago

Here's a fun problem I came across. During an hour two independent events can happen at any time. What's the probability that the events are at least 10 minutes apart?

  • This Question is Closed
  1. rational
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    |dw:1432999492312:dw|

  2. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Hahaha yeah you got it.

  3. rational
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    P(|E1-E2| > 10 minutes) = (Area of shaded region) / (Total area) = 5^2/6^2 = 25/36 ?

  4. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Yep, exactly. =)

  5. rational
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    these are really fun xD i got some practice a couple of months ago when @amistre64 was preparing for some probability theory exam

  6. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Something that just occurred to me that I don't know the answer to is what is the probability of having 3 events 10 minutes apart? Or, is there a good reason why the probability is 5^2/6^2? If the events were separated by 20 minutes would the answer then become 4^2/6^2 ? I guess this would be easy to check for the general answer.

  7. rational
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    very interesting, 3 events requires a triple integral is it

  8. rational
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 3

    second part of the question is easy yeah we always get squares in top and bottom because of symmetry : |E1-E2| > 20

  9. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    for the 2D case in general the answer is: \[\frac{(60-t)^2}{60^2}\] which is quite nice!

  10. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Wow, that's beautiful.

  11. ParthKohli
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Looks like my book has this... is this what is called "infinitistic probability"?

  12. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Yup, I guess? I guess continuous probability distribution maybe? It's important for quantum mechanics!

  13. ikram002p
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    :)

  14. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433020038430:dw|

  15. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh greater so 5/6

  16. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433020148264:dw|

  17. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is that it?

  18. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i see that for one case its, 5/6 then therefore for every other point its always 5/6 of the total there so 5/6 all the time :)

  19. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i solved it by looking at a number line

  20. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433020285038:dw|

  21. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so i thought about summing all these individual events up (kind of like an integral) however its constant here its always 5/6 of the smaller case, that event 1 happens at the one single point

  22. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Hence 5/6 for the complete thing

  23. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    here is also something interesting... prolly not so much but kinda cool to visualze lol

  24. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    cause its not good to think ofa number line but a circular number like for this or like a clock

  25. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433020557004:dw|

  26. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433020604712:dw|

  27. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    the problem is your answer seems to be wrong, the answer is 25/36 not 5/6 for the first question

  28. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ya i got that i change it xD

  29. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    for a sec i thought u were asking less than 10

  30. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh dang really

  31. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    why is it 5^/6^2?

  32. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lemmee think

  33. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    this picture: |dw:1433020730099:dw| imagine each point is when two events happen, so all the points on the diagonal happen simultaneously. so for instance when one event happens, (10,20) at that point, that will fall in the black, so it's greater or equal to 10 min right? So the ratio of the black region to the entire square is the probability.

  34. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh :O i get it

  35. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    theres a problem here with assuming the circular thing too, because once event 1 occurs later

  36. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the prob is greater for 10 mins

  37. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    :D

  38. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    or wait no, the even can happen back in time too

  39. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    This is pretty similar to the Buffon needle problem which is a cool way to approximate \(\pi\).

  40. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i dont think i understand the graph

  41. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Honestly I don't think it gets much better than this general case: \[\frac{(60-t)^2}{60^2}\] this tells you the probability that two random events happen with at least a separation of time t within an hour. Like if t=10 in the original then we get: \[\frac{(60-10)^2}{60^2} = \frac{25}{36}\] and of course the probability that they're 0 minutes apart will give us 100% and probability they are 60 min apart will give us 0% just like we expect, it's just from calculating it based off of a variable triangle size inside cut out basically lol.

  42. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    |dw:1433021207310:dw| Here are some random points picked in there. The first represents some time that one event happened and the other represents the time the second event happened. So look along the diagonal, (0,0) means both events happened at hte beginning of the our. (30,30) means both events happened at the 30 minute mark. Makes sense? Now look at the point (10,50) that means the first event happened at 10 minutes and the other event happened at 50 minutes. This is one of the points that falls within "both events happening 10 minutes apart or greater" since these are obviously separated by 40 minutes yeah?

  43. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ah i gotcha now xD

  44. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    the graph labelling didnt quite make sense to me, now its clear thanks

  45. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    cool so what if we have 3 events separated by 10 minutes, what does that cube look like?

  46. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433021555908:dw|

  47. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    You would think so, that was my first guess too, but it doesn't seem to be right if you think of evaluating say, this point: |dw:1433021616607:dw|

  48. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok lemme see more algrebrically first a+b+c<60*3 and diff between a,b,c is greater than 10

  49. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    (5/6)^3?

  50. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ^ thats a guess

  51. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    wait what is the question?is it all 3 have to be apart by 10? or as long as all 3 dont happen in 10

  52. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    |dw:1433021690423:dw|

  53. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok wat u drew there looks like its as long all 3 dont happen within 10, so u can have 2 with in 10 and other out

  54. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    So one example of an event can be like (10, 20, 30) since they are all separated by 10 min.

  55. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433021882772:dw|

  56. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    yeah what I drew is wrong I think idk haha

  57. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Yeah I just didn't wanna draw it with those lines in there cause it looked tacky

  58. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433021915541:dw|

  59. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lol

  60. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh u can think of the 2by 2 case as 1 big event again

  61. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1433022504943:dw|

  62. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so u solve for cases, when event 1 and 2 are close, and when they are not

  63. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    eh nvm i feel like there is something easier to compute off the differences

  64. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    did u already solve it?

  65. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    Nahhhh

  66. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ok :) im on it then sir

  67. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    some useful things: \( 20 \le x+y+z \le 40\) \( 10 \le x+y \le 50\) \( 10 \le x+z \le 50\) \( 10 \le y+z \le 50\)

  68. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    that what i was lookign at

  69. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    like can we write out system of equationa for lines

  70. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    and break it down into 4 or 6 or 9 cases or something

  71. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    I think this is enough to make up an integral, those are our bounds

  72. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    hey btw the first lines ganeshew drew was

  73. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    y=x+10 and x=y+10 right

  74. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    yeah

  75. Kainui
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 2

    notice they're symmetric so we only needed to find one part of that, then multiply by 2

  76. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    ugh its gonna into these inequality cases