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# Probability question :)

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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
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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
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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
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I'm curious too.

Jul 28, 2020
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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
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Try the following reasoning:

Jul 29, 2020
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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
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Thank you Alan!!! :)

iamhappy  Jul 29, 2020
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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
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Thanks Alan ...  but I do not know how to calculate that sum to get 3 either.

Melody  Jul 29, 2020
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Yeah, and because of that, I am quite confused how to find e2

iamhappy  Jul 29, 2020
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I think e2 is just e1 +1

Melody  Jul 30, 2020
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Ok.  Here is one way to do it:

Alan  Jul 30, 2020
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oh no! Alan, it is blocked by moderator! :(

iamhappy  Jul 30, 2020
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Scroll down the page labelled Answers.  You can see it there.

Alan  Jul 30, 2020
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Hi

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

Guest Jul 31, 2020
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$$\lambda$$ is the Greek letter lambda.

Alan  Jul 31, 2020
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Thanks very much Alan,

I am never quite confident about expected values.  :)

Jul 29, 2020
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Here is a picture of Alan's answer:  Sorry, it is a bit small ....

Jul 31, 2020