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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?
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?

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rational
 one year ago
Best ResponseYou've already chosen the best response.3dw:1432999492312:dw

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Hahaha yeah you got it.

rational
 one year ago
Best ResponseYou've already chosen the best response.3P(E1E2 > 10 minutes) = (Area of shaded region) / (Total area) = 5^2/6^2 = 25/36 ?

rational
 one year ago
Best ResponseYou've already chosen the best response.3these are really fun xD i got some practice a couple of months ago when @amistre64 was preparing for some probability theory exam

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Something 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.

rational
 one year ago
Best ResponseYou've already chosen the best response.3very interesting, 3 events requires a triple integral is it

rational
 one year ago
Best ResponseYou've already chosen the best response.3second part of the question is easy yeah we always get squares in top and bottom because of symmetry : E1E2 > 20

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2for the 2D case in general the answer is: \[\frac{(60t)^2}{60^2}\] which is quite nice!

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0Wow, that's beautiful.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.0Looks like my book has this... is this what is called "infinitistic probability"?

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Yup, I guess? I guess continuous probability distribution maybe? It's important for quantum mechanics!

dan815
 one year ago
Best ResponseYou've already chosen the best response.0i 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 :)

dan815
 one year ago
Best ResponseYou've already chosen the best response.0i solved it by looking at a number line

dan815
 one year ago
Best ResponseYou've already chosen the best response.0so 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

dan815
 one year ago
Best ResponseYou've already chosen the best response.0Hence 5/6 for the complete thing

dan815
 one year ago
Best ResponseYou've already chosen the best response.0here is also something interesting... prolly not so much but kinda cool to visualze lol

dan815
 one year ago
Best ResponseYou've already chosen the best response.0cause its not good to think ofa number line but a circular number like for this or like a clock

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2the problem is your answer seems to be wrong, the answer is 25/36 not 5/6 for the first question

dan815
 one year ago
Best ResponseYou've already chosen the best response.0ya i got that i change it xD

dan815
 one year ago
Best ResponseYou've already chosen the best response.0for a sec i thought u were asking less than 10

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2this 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.

dan815
 one year ago
Best ResponseYou've already chosen the best response.0theres a problem here with assuming the circular thing too, because once event 1 occurs later

dan815
 one year ago
Best ResponseYou've already chosen the best response.0the prob is greater for 10 mins

dan815
 one year ago
Best ResponseYou've already chosen the best response.0or wait no, the even can happen back in time too

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2This is pretty similar to the Buffon needle problem which is a cool way to approximate \(\pi\).

dan815
 one year ago
Best ResponseYou've already chosen the best response.0i dont think i understand the graph

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Honestly I don't think it gets much better than this general case: \[\frac{(60t)^2}{60^2}\] 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: \[\frac{(6010)^2}{60^2} = \frac{25}{36}\] 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.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2dw: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?

dan815
 one year ago
Best ResponseYou've already chosen the best response.0the graph labelling didnt quite make sense to me, now its clear thanks

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2cool so what if we have 3 events separated by 10 minutes, what does that cube look like?

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2You 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

dan815
 one year ago
Best ResponseYou've already chosen the best response.0ok lemme see more algrebrically first a+b+c<60*3 and diff between a,b,c is greater than 10

dan815
 one year ago
Best ResponseYou've already chosen the best response.0wait what is the question?is it all 3 have to be apart by 10? or as long as all 3 dont happen in 10

dan815
 one year ago
Best ResponseYou've already chosen the best response.0ok 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

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2So one example of an event can be like (10, 20, 30) since they are all separated by 10 min.

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2yeah what I drew is wrong I think idk haha

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2Yeah I just didn't wanna draw it with those lines in there cause it looked tacky

dan815
 one year ago
Best ResponseYou've already chosen the best response.0oh u can think of the 2by 2 case as 1 big event again

dan815
 one year ago
Best ResponseYou've already chosen the best response.0so u solve for cases, when event 1 and 2 are close, and when they are not

dan815
 one year ago
Best ResponseYou've already chosen the best response.0eh nvm i feel like there is something easier to compute off the differences

dan815
 one year ago
Best ResponseYou've already chosen the best response.0did u already solve it?

dan815
 one year ago
Best ResponseYou've already chosen the best response.0ok :) im on it then sir

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2some useful things: \( 20 \le x+y+z \le 40\) \( 10 \le x+y \le 50\) \( 10 \le x+z \le 50\) \( 10 \le y+z \le 50\)

dan815
 one year ago
Best ResponseYou've already chosen the best response.0that what i was lookign at

dan815
 one year ago
Best ResponseYou've already chosen the best response.0like can we write out system of equationa for lines

dan815
 one year ago
Best ResponseYou've already chosen the best response.0and break it down into 4 or 6 or 9 cases or something

Kainui
 one year ago
Best ResponseYou've already chosen the best response.2I think this is enough to make up an integral, those are our bounds

dan815
 one year ago
Best ResponseYou've already chosen the best response.0hey btw the first lines ganeshew drew was

dan815
 one year ago
Best ResponseYou've already chosen the best response.0y=x+10 and x=y+10 right

Kainui<
 one year ago
Best ResponseYou've already chosen the best response.2notice they're symmetric so we only needed to find one part of that, then multiply by 2