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March 3rd, 2015, 02:46 PM   #1
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Probabilities and Expected Values For Pairs Of Pictures

There are eight pictures that each appear twice in a 4x4 grid:

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16

There are 12 pairs of spaces that are horizontally adjacent:

1 and 2
2 and 3
3 and 4
5 and 6
6 and 7
7 and 8
9 and 10
10 and 11
11 and 12
13 and 14
14 and 15
15 and 16

1. How many arrangements of pictures are possible? It doesn't matter which picture is which within a pair, and it doesn't matter which pair is in which place. An arrangement with Picture A in spaces 1 and 2 and Picture B in spaces 3 and 4 is the same as having Picture B in spaces 1 and 2 and Picture A in spaces 3 and 4.

2. What is the expected number of pairs of horizontally adjacent spaces that have the same picture?

3. What is the probability that 0, 1, 2, 3, 4, 5, 6, 7, and 8 of the pairs of horizontally adjacent spaces will have the same picture?
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