math extra credit?

Question

An 8 foot by 8 foot area has been tiled with 1 foot square tiles. Two of the tiles were defective. What is the probability that the two defective tiles share an edge?

Answer 1

If you have an 8x8 tiled area, there are 4 corner tiles, 24 side tiles, and 36 center tiles. We'll label one defectective tiles A, and the other B, then figure out how many ways A can be next to B.

If A is a corner tile, there are only two ways that B could be next to a corner tile.
If A is a side tile, there are only three ways that B could be next to a side tile.
If A is a center tile, there are four ways that B could be next to a center tile.

If we add up all the ways that A can be placed with all the ways that B can be placed next to it we get...

4*2+24*3+36*4 = 224

Because A and B are interchangeable, we really have double the number we need. Both AB and BA were counted. So, there are 112 ways for 2 defective tiles to be next to each other in a grid.

Now let's consider all the ways that A and B can be placed randomly. There are 64 places for A, then 63 places for B.

64*63 = 4032

Again divide by 2 because A and B are not differentiable. So, there are 2016 ways to have two defective tiles in an 8x8 grid.

Therefore, the chance of A and B being next to each other is:

112 / 2016 = 1 / 18 = 0.055556 = 5.56% chance that the two tiles will share a side.

Answer 2

You have 64 tiles altogether. Call your defective tiles A and B. You must consider three cases:

A. Tile A is a corner tile (4 out of 64), which has tiles touching 2 of its edges;
B. Tile A is a side tile (24 out of 64), with tiles touching 3 of its edges;
C. Tile A is a center tile (36 out of 64), with tiles touching all 4 edges.

Since any of these cases satisfy the result you're looking for, you add their probabilities together. The probability for case A is as follows:(4/64)*(2/63)
4/64 represents the probability that a randomly chosen tile A is in the corner, and the probability that tile B will be one of the two tiles that shares an edge with that tile A. Similarly, the probability of case B is (24/64)*(3/63) and case C is (36/64)*(4/63).

Add them together to get the probability for all cases:
[(4/64)*(2/63)]+[(24/64*3/63)]+[(36/64)*(4/63)]
=28/504=1/18

Answer 3

there are 256 edges. . The probability that the 2 bad tiles share an edge is 1/255.

Answer 4

Number of ways of arranging the tiles = 64!. number of ways of arranging the defective tiles together = (32+56)*2=88*2 therefore ans= 196/64!