Calculate the expected distance that the boat travels.

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Its a discrete probability question.

Imagine a boat and two banks on an integer line.
One of the banks is at the origin and the other bank can be at any integer in range $[B+1,2B]$

If the right bank is at distance $d$ from the origin, then the boat can be at any integer point in the river in the range $[1,d-1]$.
The boat always goes towards the nearest bank.

What is the expected distance travelled by the boat?

The right bank can be any of the possible distances with equal probability and the boat can be at any of the possible feasible distances with equal probability.

Suppose the distance between two banks is $2D$, then the expected distance travelled by the boat is $(1+2ldots +D)/D=(D+1)/2$

probability-theory expectation

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edited Jul 22 at 10:47

asked Jul 21 at 14:11

Vk1

147

• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, suppose $B = 3.$ Let $(x,y)$ signify that the distance between banks is $x$ and the boat is $y$ distance from the left bank. When you say the cases are equiprobable, do you mean that $(4,1)$ and $(6,1)$ have the same probability? Or do you mean that the distance between banks is equally likely to be $4$ or $6$? Also, can the distance of the boat from a bank be $0$ (an integer) or must it be at least $1$?
  – David K
  Jul 21 at 14:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What I mean is the distance between two banks can be any integer in range of $B+1$ and $2B$ with equal probability and when the two banks are $D$ distance apart, the boat has equal probability to be at any distance from 0 to D from the left bank. If it helps you can assume that the distance is at least 1, but that may not be an important detail
  – Vk1
  Jul 21 at 14:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have changed to question for clarity, hope that helps. Yes I meant that distance between banks is equally likely to be 4 or 6.
  – Vk1
  Jul 21 at 18:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  People may be waiting to see what you tried. Supposing you knew the distance between banks was $D$ (not random), could you find the expected distance traveled?
  – David K
  Jul 21 at 23:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be about $D/4$, since best case the distance is 0 and worst case the distance is $D/2$
  – Vk1
  Jul 22 at 10:48
  

  
  
  
  

 | 
show 1 more comment

up vote
0
down vote

favorite

Its a discrete probability question.

Imagine a boat and two banks on an integer line.
One of the banks is at the origin and the other bank can be at any integer in range $[B+1,2B]$

If the right bank is at distance $d$ from the origin, then the boat can be at any integer point in the river in the range $[1,d-1]$.
The boat always goes towards the nearest bank.

What is the expected distance travelled by the boat?

The right bank can be any of the possible distances with equal probability and the boat can be at any of the possible feasible distances with equal probability.

Suppose the distance between two banks is $2D$, then the expected distance travelled by the boat is $(1+2ldots +D)/D=(D+1)/2$

probability-theory expectation

share|cite|improve this question

edited Jul 22 at 10:47

asked Jul 21 at 14:11

Vk1

147

• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, suppose $B = 3.$ Let $(x,y)$ signify that the distance between banks is $x$ and the boat is $y$ distance from the left bank. When you say the cases are equiprobable, do you mean that $(4,1)$ and $(6,1)$ have the same probability? Or do you mean that the distance between banks is equally likely to be $4$ or $6$? Also, can the distance of the boat from a bank be $0$ (an integer) or must it be at least $1$?
  – David K
  Jul 21 at 14:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What I mean is the distance between two banks can be any integer in range of $B+1$ and $2B$ with equal probability and when the two banks are $D$ distance apart, the boat has equal probability to be at any distance from 0 to D from the left bank. If it helps you can assume that the distance is at least 1, but that may not be an important detail
  – Vk1
  Jul 21 at 14:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have changed to question for clarity, hope that helps. Yes I meant that distance between banks is equally likely to be 4 or 6.
  – Vk1
  Jul 21 at 18:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  People may be waiting to see what you tried. Supposing you knew the distance between banks was $D$ (not random), could you find the expected distance traveled?
  – David K
  Jul 21 at 23:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be about $D/4$, since best case the distance is 0 and worst case the distance is $D/2$
  – Vk1
  Jul 22 at 10:48
  

  
  
  
  

 | 
show 1 more comment

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Its a discrete probability question.

Imagine a boat and two banks on an integer line.
One of the banks is at the origin and the other bank can be at any integer in range $[B+1,2B]$

If the right bank is at distance $d$ from the origin, then the boat can be at any integer point in the river in the range $[1,d-1]$.
The boat always goes towards the nearest bank.

What is the expected distance travelled by the boat?

The right bank can be any of the possible distances with equal probability and the boat can be at any of the possible feasible distances with equal probability.

Suppose the distance between two banks is $2D$, then the expected distance travelled by the boat is $(1+2ldots +D)/D=(D+1)/2$

probability-theory expectation

share|cite|improve this question

edited Jul 22 at 10:47

asked Jul 21 at 14:11

Vk1

147

Its a discrete probability question.

Imagine a boat and two banks on an integer line.
One of the banks is at the origin and the other bank can be at any integer in range $[B+1,2B]$

If the right bank is at distance $d$ from the origin, then the boat can be at any integer point in the river in the range $[1,d-1]$.
The boat always goes towards the nearest bank.

What is the expected distance travelled by the boat?

The right bank can be any of the possible distances with equal probability and the boat can be at any of the possible feasible distances with equal probability.

Suppose the distance between two banks is $2D$, then the expected distance travelled by the boat is $(1+2ldots +D)/D=(D+1)/2$

probability-theory expectation

share|cite|improve this question

edited Jul 22 at 10:47

asked Jul 21 at 14:11

Vk1

147

share|cite|improve this question

share|cite|improve this question

edited Jul 22 at 10:47

edited Jul 22 at 10:47

edited Jul 22 at 10:47

asked Jul 21 at 14:11

Vk1

147

asked Jul 21 at 14:11

Vk1

147

asked Jul 21 at 14:11

Vk1

147

147

• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, suppose $B = 3.$ Let $(x,y)$ signify that the distance between banks is $x$ and the boat is $y$ distance from the left bank. When you say the cases are equiprobable, do you mean that $(4,1)$ and $(6,1)$ have the same probability? Or do you mean that the distance between banks is equally likely to be $4$ or $6$? Also, can the distance of the boat from a bank be $0$ (an integer) or must it be at least $1$?
  – David K
  Jul 21 at 14:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What I mean is the distance between two banks can be any integer in range of $B+1$ and $2B$ with equal probability and when the two banks are $D$ distance apart, the boat has equal probability to be at any distance from 0 to D from the left bank. If it helps you can assume that the distance is at least 1, but that may not be an important detail
  – Vk1
  Jul 21 at 14:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have changed to question for clarity, hope that helps. Yes I meant that distance between banks is equally likely to be 4 or 6.
  – Vk1
  Jul 21 at 18:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  People may be waiting to see what you tried. Supposing you knew the distance between banks was $D$ (not random), could you find the expected distance traveled?
  – David K
  Jul 21 at 23:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be about $D/4$, since best case the distance is 0 and worst case the distance is $D/2$
  – Vk1
  Jul 22 at 10:48
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, suppose $B = 3.$ Let $(x,y)$ signify that the distance between banks is $x$ and the boat is $y$ distance from the left bank. When you say the cases are equiprobable, do you mean that $(4,1)$ and $(6,1)$ have the same probability? Or do you mean that the distance between banks is equally likely to be $4$ or $6$? Also, can the distance of the boat from a bank be $0$ (an integer) or must it be at least $1$?
  – David K
  Jul 21 at 14:41
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  What I mean is the distance between two banks can be any integer in range of $B+1$ and $2B$ with equal probability and when the two banks are $D$ distance apart, the boat has equal probability to be at any distance from 0 to D from the left bank. If it helps you can assume that the distance is at least 1, but that may not be an important detail
  – Vk1
  Jul 21 at 14:54
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  I have changed to question for clarity, hope that helps. Yes I meant that distance between banks is equally likely to be 4 or 6.
  – Vk1
  Jul 21 at 18:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  People may be waiting to see what you tried. Supposing you knew the distance between banks was $D$ (not random), could you find the expected distance traveled?
  – David K
  Jul 21 at 23:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It should be about $D/4$, since best case the distance is 0 and worst case the distance is $D/2$
  – Vk1
  Jul 22 at 10:48
  

  
  
  
  

OK, suppose $B = 3.$ Let $(x,y)$ signify that the distance between banks is $x$ and the boat is $y$ distance from the left bank. When you say the cases are equiprobable, do you mean that $(4,1)$ and $(6,1)$ have the same probability? Or do you mean that the distance between banks is equally likely to be $4$ or $6$? Also, can the distance of the boat from a bank be $0$ (an integer) or must it be at least $1$?
– David K
Jul 21 at 14:41

OK, suppose $B = 3.$ Let $(x,y)$ signify that the distance between banks is $x$ and the boat is $y$ distance from the left bank. When you say the cases are equiprobable, do you mean that $(4,1)$ and $(6,1)$ have the same probability? Or do you mean that the distance between banks is equally likely to be $4$ or $6$? Also, can the distance of the boat from a bank be $0$ (an integer) or must it be at least $1$?
– David K
Jul 21 at 14:41

What I mean is the distance between two banks can be any integer in range of $B+1$ and $2B$ with equal probability and when the two banks are $D$ distance apart, the boat has equal probability to be at any distance from 0 to D from the left bank. If it helps you can assume that the distance is at least 1, but that may not be an important detail
– Vk1
Jul 21 at 14:54

What I mean is the distance between two banks can be any integer in range of $B+1$ and $2B$ with equal probability and when the two banks are $D$ distance apart, the boat has equal probability to be at any distance from 0 to D from the left bank. If it helps you can assume that the distance is at least 1, but that may not be an important detail
– Vk1
Jul 21 at 14:54

I have changed to question for clarity, hope that helps. Yes I meant that distance between banks is equally likely to be 4 or 6.
– Vk1
Jul 21 at 18:31

I have changed to question for clarity, hope that helps. Yes I meant that distance between banks is equally likely to be 4 or 6.
– Vk1
Jul 21 at 18:31

People may be waiting to see what you tried. Supposing you knew the distance between banks was $D$ (not random), could you find the expected distance traveled?
– David K
Jul 21 at 23:19

People may be waiting to see what you tried. Supposing you knew the distance between banks was $D$ (not random), could you find the expected distance traveled?
– David K
Jul 21 at 23:19

It should be about $D/4$, since best case the distance is 0 and worst case the distance is $D/2$
– Vk1
Jul 22 at 10:48

It should be about $D/4$, since best case the distance is 0 and worst case the distance is $D/2$
– Vk1
Jul 22 at 10:48

 | 
show 1 more comment

1 Answer
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Let $X$ be the boat's initial location, then $X$ is uniformly distributed over $1,2,ldots,d-1$. The distance the boat travels is
$$
Y := minX, d-X.
$$
Now, $X<d-X$ iff $X<d/2$, and since $X$ and $d$ are integer, this is equivalent to $Xleqslant lfloor d/2rfloor$. It follows that
beginalign
mathbb E[Y] &= sum_j=1^d-1 (jwedge d-j)mathbb P(X=j)\
&= frac1d-1left(sum_j=1^lfloor d/2rfloor j + sum_j=lfloor d/2rfloor+1^d-1 (d-j) right)\
&= frac d2 -leftlfloorfrac d2rightrfloor left(1 - fracleftlfloorfrac d2rightrfloord-1 right) .
endalign

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answered Jul 24 at 3:35

Math1000

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1 Answer
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1 Answer
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Let $X$ be the boat's initial location, then $X$ is uniformly distributed over $1,2,ldots,d-1$. The distance the boat travels is
$$
Y := minX, d-X.
$$
Now, $X<d-X$ iff $X<d/2$, and since $X$ and $d$ are integer, this is equivalent to $Xleqslant lfloor d/2rfloor$. It follows that
beginalign
mathbb E[Y] &= sum_j=1^d-1 (jwedge d-j)mathbb P(X=j)\
&= frac1d-1left(sum_j=1^lfloor d/2rfloor j + sum_j=lfloor d/2rfloor+1^d-1 (d-j) right)\
&= frac d2 -leftlfloorfrac d2rightrfloor left(1 - fracleftlfloorfrac d2rightrfloord-1 right) .
endalign

share|cite|improve this answer

answered Jul 24 at 3:35

Math1000

18.4k31544

add a comment | 

up vote
0
down vote

Let $X$ be the boat's initial location, then $X$ is uniformly distributed over $1,2,ldots,d-1$. The distance the boat travels is
$$
Y := minX, d-X.
$$
Now, $X<d-X$ iff $X<d/2$, and since $X$ and $d$ are integer, this is equivalent to $Xleqslant lfloor d/2rfloor$. It follows that
beginalign
mathbb E[Y] &= sum_j=1^d-1 (jwedge d-j)mathbb P(X=j)\
&= frac1d-1left(sum_j=1^lfloor d/2rfloor j + sum_j=lfloor d/2rfloor+1^d-1 (d-j) right)\
&= frac d2 -leftlfloorfrac d2rightrfloor left(1 - fracleftlfloorfrac d2rightrfloord-1 right) .
endalign

share|cite|improve this answer

answered Jul 24 at 3:35

Math1000

18.4k31544

add a comment | 

up vote
0
down vote

up vote
0
down vote

Let $X$ be the boat's initial location, then $X$ is uniformly distributed over $1,2,ldots,d-1$. The distance the boat travels is
$$
Y := minX, d-X.
$$
Now, $X<d-X$ iff $X<d/2$, and since $X$ and $d$ are integer, this is equivalent to $Xleqslant lfloor d/2rfloor$. It follows that
beginalign
mathbb E[Y] &= sum_j=1^d-1 (jwedge d-j)mathbb P(X=j)\
&= frac1d-1left(sum_j=1^lfloor d/2rfloor j + sum_j=lfloor d/2rfloor+1^d-1 (d-j) right)\
&= frac d2 -leftlfloorfrac d2rightrfloor left(1 - fracleftlfloorfrac d2rightrfloord-1 right) .
endalign

share|cite|improve this answer

answered Jul 24 at 3:35

Math1000

18.4k31544

Let $X$ be the boat's initial location, then $X$ is uniformly distributed over $1,2,ldots,d-1$. The distance the boat travels is
$$
Y := minX, d-X.
$$
Now, $X<d-X$ iff $X<d/2$, and since $X$ and $d$ are integer, this is equivalent to $Xleqslant lfloor d/2rfloor$. It follows that
beginalign
mathbb E[Y] &= sum_j=1^d-1 (jwedge d-j)mathbb P(X=j)\
&= frac1d-1left(sum_j=1^lfloor d/2rfloor j + sum_j=lfloor d/2rfloor+1^d-1 (d-j) right)\
&= frac d2 -leftlfloorfrac d2rightrfloor left(1 - fracleftlfloorfrac d2rightrfloord-1 right) .
endalign

share|cite|improve this answer

answered Jul 24 at 3:35

Math1000

18.4k31544

share|cite|improve this answer

share|cite|improve this answer

answered Jul 24 at 3:35

Math1000

18.4k31544

answered Jul 24 at 3:35

Math1000

18.4k31544

answered Jul 24 at 3:35

Math1000

18.4k31544

18.4k31544

add a comment | 

add a comment | 

 

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