Two floors in a 15 -story building (numbered 1 to 15 ) are chosen independently and uniformly at random for a fire safety inspection. The same floor may be chosen twice. The probability that the two selected floors are at least 5 floors apart is :

Step-by-step solution

Let the chosen floors be x and y . Since they are chosen independently with replacement from 1, 2, , 15 , the total number of outcomes is 15 15 = 225 . We need the probability that |x - y| 5 . Case 1: x - y 5 If y = 1 , x can be 6, 7, , 15 ( 10 outcomes). If y = 2 , x can be 7, 8, , 15 ( 9 outcomes). If y = 10 , x = 15 ( 1 outcome). Number of favorable outcomes for this case is 10 + 9 + + 1 = 55 . Case 2: y - x 5 By symmetry, the number of favorable outcomes is also 55 . Total favorable outcomes = 55 + 55 = 110 . R