What is the probability of a person moving outside a circle, given only speed and time?
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We are given R (radius of circle), u (speed of person) and t (time traveled by person). Hence, we can calculate the distance traveled, D.
Assume that the starting position is evenly distributed in the circle.
Also, the person can take any direction, and the direction will be steady.
Is it possible for this to be solved for R,u and t? I can only find a solution which includes D_start, the initial distance of the person from the center of the circle.
probability geometry probability-theory circle
asked Nov 23 at 15:40
XuUserAC
543
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up vote
1
down vote
favorite
We are given R (radius of circle), u (speed of person) and t (time traveled by person). Hence, we can calculate the distance traveled, D.
Assume that the starting position is evenly distributed in the circle.
Also, the person can take any direction, and the direction will be steady.
Is it possible for this to be solved for R,u and t? I can only find a solution which includes D_start, the initial distance of the person from the center of the circle.
probability geometry probability-theory circle
asked Nov 23 at 15:40
XuUserAC
543
If you post the solution you have that includes "D_start" it will be easier for someone to help you.
– John Douma
Nov 23 at 15:47
It might be worth distinguishing disk and circle: you start on a disk and move to a point on the edge of a circle centred on your starting point
– Henry
Nov 23 at 16:03
What is your question? The expected value for time to get to the circle?
– Moti
Nov 23 at 20:33
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
We are given R (radius of circle), u (speed of person) and t (time traveled by person). Hence, we can calculate the distance traveled, D.
Assume that the starting position is evenly distributed in the circle.
Also, the person can take any direction, and the direction will be steady.
Is it possible for this to be solved for R,u and t? I can only find a solution which includes D_start, the initial distance of the person from the center of the circle.
probability geometry probability-theory circle
asked Nov 23 at 15:40
XuUserAC
543
We are given R (radius of circle), u (speed of person) and t (time traveled by person). Hence, we can calculate the distance traveled, D.
Assume that the starting position is evenly distributed in the circle.
Also, the person can take any direction, and the direction will be steady.
Is it possible for this to be solved for R,u and t? I can only find a solution which includes D_start, the initial distance of the person from the center of the circle.
probability geometry probability-theory circle
probability geometry probability-theory circle
asked Nov 23 at 15:40
XuUserAC
543
asked Nov 23 at 15:40
XuUserAC
543
asked Nov 23 at 15:40
XuUserAC
543
asked Nov 23 at 15:40
XuUserAC
543
asked Nov 23 at 15:40
XuUserAC
543
543
If you post the solution you have that includes "D_start" it will be easier for someone to help you.
– John Douma
Nov 23 at 15:47
It might be worth distinguishing disk and circle: you start on a disk and move to a point on the edge of a circle centred on your starting point
– Henry
Nov 23 at 16:03
What is your question? The expected value for time to get to the circle?
– Moti
Nov 23 at 20:33
add a comment |
If you post the solution you have that includes "D_start" it will be easier for someone to help you.
– John Douma
Nov 23 at 15:47
It might be worth distinguishing disk and circle: you start on a disk and move to a point on the edge of a circle centred on your starting point
– Henry
Nov 23 at 16:03
What is your question? The expected value for time to get to the circle?
– Moti
Nov 23 at 20:33
If you post the solution you have that includes "D_start" it will be easier for someone to help you.
– John Douma
Nov 23 at 15:47
If you post the solution you have that includes "D_start" it will be easier for someone to help you.
– John Douma
Nov 23 at 15:47
It might be worth distinguishing disk and circle: you start on a disk and move to a point on the edge of a circle centred on your starting point
– Henry
Nov 23 at 16:03
It might be worth distinguishing disk and circle: you start on a disk and move to a point on the edge of a circle centred on your starting point
– Henry
Nov 23 at 16:03
What is your question? The expected value for time to get to the circle?
– Moti
Nov 23 at 20:33
What is your question? The expected value for time to get to the circle?
– Moti
Nov 23 at 20:33
add a comment |
1 Answer
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Fix the disk radius $R=1$ without loss of generality.
I selected the speed $u=1$—traveling a unit distance in $1$ second—just as an example.
Then, on average, it takes about $0.85$ seconds
for a random start point, aiming in a random direction, to exit the disk.
So for $u=1,,t=0.85$, the probability of exiting the disk is $50$%.
$50$ random unit-length rays within a unit-radius disk.
For an arbitrary speed $u>0$, it takes about $0.85/u$ on average
to escape the disk.
However, I do not know an analytical expression for the $0.85$ constant.
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Fix the disk radius $R=1$ without loss of generality.
I selected the speed $u=1$—traveling a unit distance in $1$ second—just as an example.
Then, on average, it takes about $0.85$ seconds
for a random start point, aiming in a random direction, to exit the disk.
So for $u=1,,t=0.85$, the probability of exiting the disk is $50$%.
$50$ random unit-length rays within a unit-radius disk.
For an arbitrary speed $u>0$, it takes about $0.85/u$ on average
to escape the disk.
However, I do not know an analytical expression for the $0.85$ constant.
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
add a comment |
up vote
0
down vote
Fix the disk radius $R=1$ without loss of generality.
I selected the speed $u=1$—traveling a unit distance in $1$ second—just as an example.
Then, on average, it takes about $0.85$ seconds
for a random start point, aiming in a random direction, to exit the disk.
So for $u=1,,t=0.85$, the probability of exiting the disk is $50$%.
$50$ random unit-length rays within a unit-radius disk.
For an arbitrary speed $u>0$, it takes about $0.85/u$ on average
to escape the disk.
However, I do not know an analytical expression for the $0.85$ constant.
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
add a comment |
up vote
0
down vote
up vote
0
down vote
Fix the disk radius $R=1$ without loss of generality.
I selected the speed $u=1$—traveling a unit distance in $1$ second—just as an example.
Then, on average, it takes about $0.85$ seconds
for a random start point, aiming in a random direction, to exit the disk.
So for $u=1,,t=0.85$, the probability of exiting the disk is $50$%.
$50$ random unit-length rays within a unit-radius disk.
For an arbitrary speed $u>0$, it takes about $0.85/u$ on average
to escape the disk.
However, I do not know an analytical expression for the $0.85$ constant.
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
Fix the disk radius $R=1$ without loss of generality.
I selected the speed $u=1$—traveling a unit distance in $1$ second—just as an example.
Then, on average, it takes about $0.85$ seconds
for a random start point, aiming in a random direction, to exit the disk.
So for $u=1,,t=0.85$, the probability of exiting the disk is $50$%.
$50$ random unit-length rays within a unit-radius disk.
For an arbitrary speed $u>0$, it takes about $0.85/u$ on average
to escape the disk.
However, I do not know an analytical expression for the $0.85$ constant.
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
edited Nov 25 at 13:25
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
answered Nov 25 at 1:44
Joseph O'Rourke
17.5k348106
17.5k348106
add a comment |
add a comment |
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If you post the solution you have that includes "D_start" it will be easier for someone to help you.
– John Douma
Nov 23 at 15:47
It might be worth distinguishing disk and circle: you start on a disk and move to a point on the edge of a circle centred on your starting point
– Henry
Nov 23 at 16:03
What is your question? The expected value for time to get to the circle?
– Moti
Nov 23 at 20:33