Theres a chess board.. regular one with 64 squares.
If two squares are selected at random Find the probability that they have a common side..
THnx In advance...!
There are 4 corner squares, out of a total of 64, therefore the probability of the first sqyuare being a corner is 1/16. Then there are 8 squares adjacent to a corner square so the probability of choosing one of thes3es as the second square is 8/63. So the probability of choosing two adjacent squares the first of which is a corner is 1/126.
Now repeat for edges and interior first choices.
CB
Hello, umangarora!
There's a chessboard ... regular one with 64 squares.
If two squares are selected at random,
find the probability that they have a common side.
Choosing 2 squares from the available 64 squares,
. . there are: .$\displaystyle _{64}C_2 \:=\:{64\choose2} \:=\:2016$ choices.
How many of these are "dominos"? . $\displaystyle \square\!\square$
In a row, $\displaystyle \square\!\square\!\square\!\square\!\square\!\squa re\!\square\!\square$, there are 7 ways to place a domino.
. . With 8 rows, there are 56 possible "horizontal" dominos.
The same is true for the columns: .56 possible "vertical" dominos.
Hence, there are: .$\displaystyle 56 + 56 \:=\:112$ possible dominos.
The probability is: .$\displaystyle \dfrac{112}{2016} \:=\:\dfrac{1}{18}$
Alternatively,
If a chosen square is in a corner, then the probabilty that the 2nd chosen square
is not sharing a side with it is
$\displaystyle \displaystyle\frac{61}{63}$
hence the probability that the squares have one of them in the corner,
with another not sharing a side is
$\displaystyle \displaystyle\frac{4}{64}\ \frac{61}{63}$
There are 6(4)=24 remaining squares along the edge that are not in a corner.
The probability of another square not sharing a side with one of these is
$\displaystyle \displaystyle\frac{60}{63}$
hence the probability that the squares include one of the remaining edge squares,
but the other does not share a side with it is
$\displaystyle \displaystyle\frac{24}{64}\ \frac{60}{63}$
There are 36 squares not on an edge.
Therefore the probability that the squares include one of these, and the 2nd one is not sharing a side with it is
$\displaystyle \displaystyle\frac{36}{64}\ \frac{59}{63}$
Therefore, the probability that 2 squares are selected and do share a side is
$\displaystyle \displaystyle\ 1-\left(\frac{4(61)+(24)(60)+(36)(59)}{(64)(63)}\rig ht)=\frac{(64)(63)-3808}{(64)(63)}=\frac{4032-3808}{4032}=\frac{224}{4032}=\frac{1}{18}$