+0  
 
+8
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avatar+359 

Fred the ant is on the real number line, and Fred is trying to get to the point 0.

If Fred is at 1 then on the next step, Fred moves to either 0 or 2 with equal probability. If Fred is at 2 then on the next step, Fred always moves to 1.

Let \(e_1\) be expected number of steps Fred takes to get to 0 given that Fred starts at the point 1. Similarly, let \(e_2\) be expected number of steps Fred takes to get to 0 given that Fred starts at the point 2.

Determine the ordered pair \((e_1, e_2)\).

 

Thanks in advance for anyone who can help! Thank you!

 Jul 26, 2020
 #1
avatar
+3

You can track the probabilities using different case, i.e. Anty gets to 0 after 1 steps, 2 steps, 3 steps, and so on.  This gives us e = 1/2*1 + 1/4*2 + 1/8*3 + 1/16*4 + ... By arithmetico-geometric series, e = 2.  Similarly, f = 1/2*2 + 1/4*3 + 1/8*4 + 1/16*5 + ... = 3, so (e,f) = (2,3).

 Jul 26, 2020
 #2
avatar+359 
+7

thank you guest for your effort and time, but I think that (2,3) may be wrong. I don't know what is wrong with it, since I think that the reasoning is quite clear, but I will take a few moments to see what could be wrong with it.

 Jul 27, 2020
 #3
avatar+111077 
0

I'm curious too.  frown

 Jul 28, 2020
 #4
avatar+359 
+6

I searched up, and this was actually posted before, and guest used the same answer both times. The other one has four thumbs down for guest which I do not clearly understand since guest must have spent a bit of effort into the answer. 

iamhappy  Jul 28, 2020
 #5
avatar+31085 
+3

Try the following reasoning:

 Jul 29, 2020
 #6
avatar+111077 
0

Alan's post is being blocked here but available on the answer page  for some weird reason

so I have taken a pit of it and will try to display it here.

 

 

 Jul 29, 2020
 #8
avatar+359 
+5

Thank you Alan!!! :)

iamhappy  Jul 29, 2020
 #9
avatar+359 
+6

Alan, can you explain more how you got from second to last step to the answer? I don't seem to quite understand. :( I do understand the beginnning and middle though. Good reasoning!!! 

iamhappy  Jul 29, 2020
 #10
avatar+111077 
0

Thanks Alan ...  but I do not know how to calculate that sum to get 3 either.  

Melody  Jul 29, 2020
 #11
avatar+359 
+5

Yeah, and because of that, I am quite confused how to find e2

iamhappy  Jul 29, 2020
 #12
avatar+111077 
+1

I think e2 is just e1 +1

Melody  Jul 30, 2020
 #13
avatar+31085 
+3

Ok.  Here is one way to do it:

 

Alan  Jul 30, 2020
 #14
avatar+359 
+5

oh no! Alan, it is blocked by moderator! :(

iamhappy  Jul 30, 2020
 #15
avatar+31085 
0

Scroll down the page labelled Answers.  You can see it there.

Alan  Jul 30, 2020
 #16
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0

Hi

If anyone can answer please, what is the concept of \(\lambda \)  in sums that Alan used called?

Guest Jul 31, 2020
 #18
avatar+31085 
0

\(\lambda\) is the Greek letter lambda.

Alan  Jul 31, 2020
 #7
avatar+111077 
+2

Thanks very much Alan,

I am never quite confident about expected values.  :)

 Jul 29, 2020
 #17
avatar+111077 
+3

Here is a picture of Alan's answer:  Sorry, it is a bit small ....

 

 Jul 31, 2020

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