Square - probability

[Vemuri]

The probability that a square selected at random from a 8X8 chess board is of size 3X3 is:

A) 8/51
B) 14/17
C) 13/17
D) 25/204
E) 3/17

[billzhao]

The probability that a square selected at random from a 8X8 chess board is of size 3X3 is:

A) 8/51
B) 14/17
C) 13/17
D) 25/204
E) 3/17

My answer is (E).

the number of square of size 1*1: 8*8=64
the number of square of size 2*2: 7*7=49
the number of square of size 3*3: 6*6=36
the number of square of size 4*4: 5*5=25
the number of square of size 5*5: 4*4=16
the number of square of size 6*6: 3*3=9
the number of square of size 7*7: 2*2=4
the number of square of size 8*8: 1*1=1

So The probability that a square selected at random from a 8X8 chess board is of size 3X3 is 36/(1+4+9+16...+64) = 3/17

[Vemuri]

My answer is (E).

the number of square of size 1*1: 8*8=64
the number of square of size 2*2: 7*7=49
the number of square of size 3*3: 6*6=36
the number of square of size 4*4: 5*5=25
the number of square of size 5*5: 4*4=16
the number of square of size 6*6: 3*3=9
the number of square of size 7*7: 2*2=4
the number of square of size 8*8: 1*1=1

So The probability that a square selected at random from a 8X8 chess board is of size 3X3 is 36/(1+4+9+16...+64) = 3/17

E is the right answer.

I am not sure I understand your solution. Can you please explain? I understood that the number of square of size 1X1 is: 8X8. But, how did you derive the others?

Appreciate your response.

[cramya]

Vemuri,
One way to understand would be to draw a 8 by 8 board.

If u pick a 2*2 square starting with the first 2 columns 1 and 2 u will get 7 new squares down the columns 1 and 2 and this can be extended to remaining columns except the last resulting in 7*7 = 49


1 2(7 squares down the columns 1 and 2)
2 3(7 squares down the columns 2 and 3)
3 4
4 5
5 6
6 7
7 8

Similarly extend the same idea for the others. Hope this helps! Let me know if u still have questions.

Good solution Bill!

Regards,
CR

[cramya]

Sometimes a picture speaks a thousand words and for this prob it doesnt take logn to draw a rough 8*8 square. Just to get the idea...

May not work(drawing part) if its a 10000*10000 square... but we can cross that bridge when and if it comes

Regards,
CR