Answer:
0.0593, or 5.93%
Explanation:
There are a total of 8 x 8 = 64 one-foot tiles in the foyer. Since two of them are cracked, there are 62 tiles that are not cracked.
To find the probability that the two cracked tiles share a common edge, we need to first determine the total number of pairs of adjacent tiles in the 62 remaining tiles. Each tile has four adjacent tiles (unless it is a tile on the edge of the foyer), so the total number of pairs of adjacent tiles is:
4 x (62 - 14) + 3 x 8 = 220
We subtracted 14 from 62 because there are 14 tiles on the edges of the foyer that only have three adjacent tiles, and we added 3 x 8 to account for the eight corner tiles that only have two adjacent tiles.
Next, we need to determine the number of ways that the two cracked tiles can be placed such that they share a common edge. There are 60 possible locations for the first cracked tile, and once it is placed, there are only 3 possible locations for the second cracked tile (it can only be placed in one of the three tiles adjacent to the first cracked tile). Therefore, the total number of ways that the two cracked tiles can be placed next to each other is:
60 x 3 = 180
Finally, we can calculate the probability by dividing the number of favorable outcomes (i.e., the number of ways that the cracked tiles can be placed next to each other) by the total number of possible outcomes (i.e., the total number of ways that the two cracked tiles can be placed):
P = 180/ (62 choose 2) = 180/ (62 x 61/2) = 0.0593 (approximately)
Therefore, the probability that the two cracked tiles share a common edge is approximately 0.0593, or 5.93%.