Question 1197385
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The analysis by tutor @ikleyn to find the number of ways to end at B is good.  But her answer to the question is flawed, because there are not always 4 choices of a direction to go from a given vertex; the total number of possible paths of length 5 <b>starting from point A on the lattice</b> is far less than 4^5.<br>
In the end, I believe the problem has a very simple solution and answer.<br>
In 5 moves on the given lattice, the only points where the ant can end up are the four points 1 unit away from A.  Point B is one of those four points; and the symmetry of the lattice means there is equal probability of ending up at any one of those four points.<br>
So the probability of ending at B after 5 moves is 1/4.<br>
ANSWER: 1/4<br>
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Response to tutor @ikleyn....<br>
What an absurd note from you regarding my response to the question!<br>
Did you take any time to read my response? or were you just reacting to the idea that anyone would dare to disagree with you?<br>
I will not agree with you when you are WRONG!<br>
The path EEEEE, <b>starting from point A on the lattice</b>, is not possible -- it goes off the lattice.  In fact, it is obvious (to anyone who is paying attention to the problem) that a path EEEEE is not possible from ANY point on the lattice.<br>
In your effort to show your superiority (?) in all things mathematical, you might try paying attention to the problem that you are working on.<br>