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An ant moves on the following lattice, beginning at the dot labeled Q . Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled B ?

 Feb 13, 2019
 #1
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0.21651

 Feb 13, 2019
 #2
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An ant moves on the following lattice, beginning at the dot labeled A.
Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random.
What is the probability that after 5 minutes he is at the dot labeled B ?

 

My attempt:

\(\begin{array}{|r|r|r|} \hline & \text{path} & \text{probability} & \text{probability} \\ \hline 1& 3,2,3,2,3,&0.0039062500 \\ 2& 3,2,3,4,3,&0.0039062500\\ 3& 3,2,3,7,3,&0.0019531250\\ 4& 3,2,3,1,3,&0.0078125000 \\ 5& 3,2,6,2,3,&0.0039062500 \\ 6& 3,2,6,7,3,&0.0019531250 \\ 7& 3,4,3,2,3,&0.0039062500 \\ 8& 3,4,3,4,3,&0.0039062500 \\ 9& 3,4,3,7,3,&0.0019531250 \\ 10& 3,4,3,1,3,&0.0078125000 \\ 11& 3,4,8,4,3,&0.0039062500 \\ 12& 3,4,8,7,3,&0.0019531250 \\ 13& 3,7,3,2,3,&0.0019531250 \\ 14& 3,7,3,4,3,&0.0019531250 \\ 15& 3,7,3,7,3,&0.0009765625 \\ 16& 3,7,3,1,3,&0.0039062500 \\ 17& 3,7,6,2,3,&0.0019531250 \\ 18& 3,7,6,7,3,&0.0009765625 \\ 19& 3,7,8,4,3,&0.0019531250 \\ 20& 3,7,8,7,3,&0.0009765625 \\ 21& 3,7,11,7,3,&0.0009765625 \\ 22& 3,1,3,2,3,&0.0078125000 \\ 23& 3,1,3,4,3,&0.0078125000 \\ 24& 3,1,3,7,3,&0.0039062500 \\ 25& 3,1,3,1,3,&0.0156250000 \\ 26& 6,2,3,2,3,&0.0039062500 \\ 27& 6,2,3,4,3,&0.0039062500 \\ 28& 6,2,3,7,3,&0.0019531250 \\ 29& 6,2,3,1,3,&0.0078125000 \\ 30& 6,2,6,2,3,&0.0039062500 \\ 31& 6,2,6,7,3,&0.0019531250 \\ 32& 6,5,6,2,3,&0.0078125000 \\ 33& 6,5,6,7,3,&0.0039062500 \\ 34& 6,7,3,2,3,&0.0019531250 \\ 35& 6,7,3,4,3,&0.0019531250 \\ 36& 6,7,3,7,3,&0.0009765625 \\ 37& 6,7,3,1,3,&0.0039062500 \\ 38& 6,7,6,2,3,&0.0019531250 \\ 39& 6,7,6,7,3,&0.0009765625 \\ 40& 6,7,8,4,3,&0.0019531250 \\ 41& 6,7,8,7,3,&0.0009765625 \\ 42& 6,7,11,7,3,&0.0009765625 \\ 43& 6,10,6,2,3,&0.0039062500 \\ 44& 6,10,6,7,3,&0.0019531250 \\ 45& 6,10,11,7,3,&0.0019531250 \\ 46& 8,4,3,2,3,&0.0039062500 \\ 47& 8,4,3,4,3,&0.0039062500 \\ 48& 8,4,3,7,3,&0.0019531250 \\ 49& 8,4,3,1,3,&0.0078125000 \\ 50& 8,4,8,4,3,&0.0039062500 \\ 51& 8,4,8,7,3,&0.0019531250 \\ 52& 8,7,3,2,3,&0.0019531250 \\ 53& 8,7,3,4,3,&0.0019531250 \\ 54& 8,7,3,7,3,&0.0009765625 \\ 55& 8,7,3,1,3,&0.0039062500 \\ 56& 8,7,6,2,3,&0.0019531250 \\ 57& 8,7,6,7,3,&0.0009765625 \\ 58& 8,7,8,4,3,&0.0019531250 \\ 59& 8,7,8,7,3,&0.0009765625 \\ 60& 8,7,11,7,3,&0.0009765625 \\ 61& 8,9,8,4,3,&0.0078125000 \\ 62& 8,9,8,7,3,&0.0039062500 \\ 63& 8,12,8,4,3,&0.0039062500 \\ 64& 8,12,8,7,3,&0.0019531250 \\ 65& 8,12,11,7,3,&0.0019531250 \\ 66& 11,7,3,2,3,&0.0019531250 \\ 67& 11,7,3,4,3,&0.0019531250 \\ 68& 11,7,3,7,3,&0.0009765625 \\ 69& 11,7,3,1,3,&0.0039062500 \\ 70& 11,7,6,2,3,&0.0019531250 \\ 71& 11,7,6,7,3,&0.0009765625 \\ 72& 11,7,8,4,3,&0.0019531250 \\ 73& 11,7,8,7,3,&0.0009765625 \\ 74& 11,7,11,7,3,&0.0009765625 \\ 75& 11,10,6,2,3,&0.0039062500 \\ 76& 11,10,6,7,3,&0.0019531250 \\ 77& 11,10,11,7,3,&0.0019531250 \\ 78& 11,12,8,4,3,&0.0039062500 \\ 79& 11,12,8,7,3,&0.0019531250 \\ 80& 11,12,11,7,3,&0.0019531250 \\ 81& 11,13,11,7,3,&0.0039062500& \text{sum } = 0.2500000000 \\ \hline \end{array}\)

 

The probability that after 5 minutes he is at the dot labeled B is \(0.25\ (25\ \%)\)

 

laugh

 Feb 13, 2019

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